Trigonometric Functions Questions

We provide trigonometric functions practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on trigonometric functions skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of trigonometric functions Questions

Question No Questions Class
1 ze
+cos 20+ cos
+…+cos (3)
10. Let f(0) =
7e
sin – +sin 20+ sin —
+…+sin (3n-2)
then
C.11 (= (2+v3) d. none of these
d. none of these
11
2 8. The minimum value of
(3 sin x – 4cos X – 10)(3 sin x + 4 cos x – 10)
11
3 22. If 0 < x < 21 and cosx | < sinx, then
then
a. the set of all values of x is
b. the number of solutions that are integral multiple of

is four
c. the sum of the largest and the smallest solution is to
TTC IT 37
d. the set of all values of x is xe
2
11
4 If ( sin theta-sqrt{6} cos theta=sqrt{7} cos theta . ) Prove that
( cos theta-sqrt{6} sin theta-sqrt{7} sin theta=1 )
11
5 9. Let a, b, and y be some angles in the first quadrant
satisfying tan(a + B) = 15/8 and cosec y = 17/8, then
which of the following hold(s) good?
a. a + B+ y = 11
b. cot a cot ßcot y=cot a + cotß+ coty
c. tan a + tanß+ tan y = tan a tan ſ tan y
d. tan a tan ß + tan ſ tan y+ tan y tan a=1
11
6 Find the value of ( frac{sin x}{1+cos x} ) at ( x=frac{pi}{4} ) 11
7 ( sum_{k=1}^{6}left[sin frac{2 k pi}{7}-i cos frac{2 k pi}{7}right]= )
A . -1
B.
( c cdot-i )
D.
11
8 32. If sin 0, – sin 02 = a and cos 6, + cos 02 = b, then
a. a? + b2 24
c. a² +6²23 d. a² +6²52
b. c? + b2 = 4
svart
11
9 20. The value of cosec
18
.
11
10 Prove the following identities. ( sin h(-x)=-sin h x ) 11
11 69. If tan 15° = 2 – 73, the value of
tan 15° cot 75° + tan 75° cot 15°
is
(2) 12
(1) 14
(3) 10
(4) 8
11
12 If ( boldsymbol{f}(sin 2 boldsymbol{x})= )
( frac{left(2 tan x+sec ^{2} xright)(1+cos 2 c)}{2}, ) then
determine the range of ( boldsymbol{f}(boldsymbol{t}) ) if range is ( [a, b], ) then ( b=? )
11
13 71. The value of sin 15° is:
22
(4) 12+ 45
750 is
11
14 If ( frac{cos (boldsymbol{A}-boldsymbol{B})}{cos (boldsymbol{A}+boldsymbol{B})}+frac{cos (boldsymbol{C}+boldsymbol{D})}{cos (boldsymbol{C}-boldsymbol{D})}=mathbf{0}, ) then
( tan A tan B tan C= )
( mathbf{A} cdot tan D )
B. ( cot D )
( c .-tan D )
D. – cot ( D )
11
15 16.
In a triangle PQR, ZR=1t/2.Iftan (P/2) and tan (0/2) ar
the roots of the equation ax² +bx+c=0 (a + 0) then.
(1999 – 2 Marks)
(a) a+b=c
(b) b+c=a
(c) a+c=b
(d) b=c
11
16 Prove that ( frac{tan mathbf{A}}{mathbf{1 + operatorname { s e c } A}}-frac{tan A}{mathbf{1 – operatorname { s e c } A}}= )
( 2 operatorname{cosec} A )
11
17 90. If no solution of 3 sin y + 12 sin’x = a lies on the line
y = 3x, then
a. a € (-o,-9) U (9,-)
b. a e [-9,9]
c. ae {-9,9}
d. none of these
11
18 m
1. If tan a =-
and tan ß =
1, find the possible
m
+1
values of (a + B).
(IIT-JEE 1978)
11
19 53. One of the general solutions of V3 cos 0 – 3 sin 0 = 4
sin 20 cos 30 is
a. mn + /18, me Z b. mtt/2 + 7t/6, y me Z
11
20 Domain of ( sin (cos theta) ) is ( ldots ldots )
A ( cdotleft[-frac{pi}{2}, frac{pi}{2}right] )
в. ( R )
c. ( [0 . pi] )
D cdot [-1,1]
11
21 11. The maximum value of y=
sinº x + cos x’
11
22 ( frac{cos theta}{sin theta} times frac{tan theta}{csc theta} ) 11
23 Find the radian measures
corresponding to ( 5^{0} 37^{prime} 0^{prime prime} )
11
24 3. in a triangle ABC, 2C –
3. In a triangle ABC, ZC =
Iftan (1) and tan ()
tan
and tan
are the roots of the equation ax² + bx + c = 0 (a + 0),
then the value of a+b (where, a, b, c are sides
of A opposite to angles A, B, C, respectively) is
11
25 Illustration 3.105 In AABC
O Prove that cos? 4 + cos2 +cose
WIt cos? At.co tcost = y(x2 + 4) then find the
maximum value of y.
11
26 If ( tan theta+tan 4 theta+tan 7 theta= )
( tan theta tan 4 theta tan 7 theta, ) then ( theta= )
A ( cdot frac{n pi}{4} )
в. ( frac{n pi}{7} )
c. ( frac{n pi}{12} )
D. ( n pi )
11
27 General solution of equation ( cot theta+ ) ( operatorname{cosec} theta=sqrt{3} ) is
A ( cdot 2 n pi+frac{pi}{4} )
B . ( (2 n-1) pi )
c. ( 2 n pi+frac{pi}{3} )
D. ( 2 n pi+frac{pi}{6} )
11
28 Prove that: ( sum sin ^{4} frac{r pi}{16}=frac{3}{2}, r= )
1,3,5,7
11
29 1. If
+ B+Y = 2੮, then
a tan tan tan ਨੂੰ = ਨੂੰ n en ?
tan
+
tan

c tan ਨੂੰ +tan : +tan ਨੂੰ –tan tan , un ਨੂੰ
d. none of these
(IIT-JEE 1979)
11
30 The area of a sector of a circle of radius
( mathbf{7} mathrm{cm} ) and central angle ( 120^{circ} ) is
A ( cdot 152 mathrm{cm}^{2} )
B. ( frac{154}{3} mathrm{cm}^{2} )
c. ( frac{128}{3} mathrm{cm}^{2} )
D. ( 128 mathrm{cm}^{2} )
11
31 Let ( [x] ) be the greatest integar function. Then the equation ( sin x=[1+sin x]+[1- )
( cos x] ) has
A . one solution in ( left[frac{-pi}{2}, frac{pi}{2}right] )
B. one solution in ( left[frac{pi}{2}, piright] )
c. one solution in R
D. no solution in R
11
32 9. cos(a-B) = 1 and cos(a+B) = 1/e, where a, ße [- TT,
Number of pairs of a, ß which satisfy both the equations
is
2.0
a. 0
c. 2
b. 1
d. 4
t
(IT-JEE 2009
(IIT-JEE 2005)
11
33 Write ( tan theta ) in terms of ( sin theta ) 11
34 Express as product :
( sin 6 x-sin 2 x )
11
35 37. If sin e, sin 02 – cose, cos e, + 1 = 0, then the value of
tan(0/2)cot(02/2) is equal to
b. 120
c. 2
d. – 2
a. – 1
11
36 2. The set of all x in the interval [0, a) for which 2 sin’x –
3 sin x + 1 20 is
(IIT-JEE 1987)
11
37 tan a + tan y
1 + tan tan
prove that
Illustration 3.63 If tan ß =
sin 2a + sin 2y
sin 2ß=
1+sin 2a sin 2y
11
38 2. If tan O= —
then sin
is
44
Lor –
a.


but not
b.
but not
d. none of these
11
39 Illustration 2.5 If (sec A + tan A) (sec B + tan B) (sec C +
tan C) = (sec A – tan A) (sec B -tan B) (sec C- tan C), prove
that the value of each side is +1.
11
40 In right angle ( triangle A B C, angle B=90^{circ}, angle A= )
( 45^{circ}=angle C, ) then ( operatorname{cosec} 45^{circ}= )
( mathbf{A} cdot mathbf{1} )
B. ( sqrt{2} )
c. ( frac{1}{sqrt{2}} )
D. None of above
11
41 ( (sin theta-sec theta)^{2}+(cos theta-operatorname{cosec} theta)^{2}= )
( (1-sec theta cdot operatorname{cosec} theta)^{2} )
11
42 Solve :
( frac{1+cos x}{sin x cos x} )
11
43 5. If in a SABC a cos” () +ccos? ()
5.
If in a AAB
COS
, then the sides
then the sides
[2003]
a, bandc
(a) satisfy a +b=C
(b) are in A.P
(c) are in G.P
(d) are in H.P.
11
44 The equation ( sin ^{2} theta-frac{4}{sin ^{3} theta-1}=1 )
( frac{4}{sin ^{3} theta-1} ) has
A. No root
B. One root
c. ( T w o ) roots
D. Infinite roots
11
45 Illustration 3.32 Prove that cos a + cos ß + cos y + cos(a +
a + B β +γ γ +α
B+ ) = 4 cos —
2
2
:
COS
COS
11
46 86. In any triangle ABC, sin A – sinB + sinʼC is always equal
to
a. 2 sin A sin B cos C b. 2 sin A cos B sin C
c. 2 sin A cos B cos C d. 2 sin A sin B sin C
11
47 [
mathrm{f}_{mathrm{f}} prod_{r=4}^{8} cos left(frac{theta}{2^{r}}right)=frac{sin left(frac{theta}{2^{n_{1}}}right)}{(2)^{n_{2}} sin left(frac{theta}{2^{n_{3}}}right)}
]
then the value of ( n_{1}+n_{3}-n_{2} ) is
11
48 and
Illustration 3.82 In triangle ABC, if cot A cot C =
cot B. cot C= =, then the value of tan C is
N-
d
11
49 If ( cos 2 B=frac{cos (A+C)}{cos (A-C)}, ) then
( tan A, tan B, tan C ) are in
A. ( A . P . )
в. ( G . P )
c. ( H . P )
D. None of these
11
50 The general solution of ( 4 tan ^{2} theta= ) ( mathbf{3} sec ^{2} boldsymbol{theta} ) is ( boldsymbol{theta}=boldsymbol{n} boldsymbol{pi} pm frac{boldsymbol{pi}}{boldsymbol{m}} . ) Then, find the
value of ( boldsymbol{m} )
11
51 10. If 2 sec A – secA – 2 cosec-A + cosec- A = 15/4, then
tan A is equal to
a. 1/2
b. 1/2
c. 1/2 √2
d. -1/2
11
52 If ( frac{3 pi}{4}<alpha<pi, ) then ( sqrt{2 cot alpha+frac{1}{sin ^{2} alpha}} )
is equal to
A. ( 1-cot alpha )
B. ( 1+cot alpha )
c. ( -1+cot alpha )
D. ( -1-cot alpha )
11
53 ( frac{tan ^{2} theta}{1+sec theta}+1 ) equals to
( A cdot tan theta )
B. ( frac{1}{cos theta} )
( mathbf{c} cdot sec theta-1 )
( mathbf{D} cdot sec theta+tan theta )
11
54 Prove that:
( cot ^{2} frac{pi}{6}+operatorname{cosec} frac{5 pi}{6}+3 tan ^{2} frac{pi}{6}=6 )
11
55 Convert ( pi / 6 ) rad to degrees. 11
56 Express the following angles into radian
( 10^{circ}, 40^{circ}, 30^{circ} )
11
57 Find the least value of ( 2 sin ^{2} theta+3 cos ^{2} theta ) 11
58 Illustration 3.58 If cos 0 = cos a cos B, prove that
ata O-a
tan
– = tan
– tan –
11
59 12. The maximum value of cos?(45° + x) + (sin x – cos x)2 is 11
60 The value of ( cos ^{2} A+cos ^{2} B- )
( mathbf{2} cos boldsymbol{A} cos boldsymbol{B} cos (boldsymbol{A}+boldsymbol{B})- )
( sin ^{2}(boldsymbol{A}+boldsymbol{B}) )
11
61 Find the radian measure of the interior
angle of regular hexagon.
11
62 If ( boldsymbol{pi}=mathbf{1 8 0}^{circ} ) and ( boldsymbol{A}=frac{boldsymbol{pi}}{boldsymbol{6}}, ) prove that
( frac{(1-cos A)(1+cos A)}{(1-sin A)(1+sin A)}=frac{1}{3} )
11
63 ( cos ^{2} alpha-sin ^{2} alpha=tan ^{2} beta . ) then show
that ( tan ^{2} alpha=cos ^{2} beta-sin ^{2} beta )
11
64 Prove that: ( tan left(11 frac{1^{circ}}{4}right)= )
( sqrt{4+2 sqrt{2}}-(sqrt{2}+1) )
11
65 In a right triangle ( Delta A B C ), right angled
at ( B . ) If ( tan A=1, ) then verify that
( 2 sin A cdot cos A=1 )
11
66 11. If p cosec 0+q cot 0 = 2 and p2 cosec? 0 – q? cot 0 = 5
then the value of 181p-2-9-2 is
11
67 For ( -frac{pi}{2}<theta<frac{pi}{2}, ) range of ( f(theta)= )
( frac{sin theta+sin 2 theta}{1+cos theta+cos 2 theta} ) is
A. ( (-infty, infty) )
a
в. (-2,2)
( c cdot(0, infty) )
D. none of these
11
68 Find the value of other five
trigonometric ratios:
( tan x=-frac{5}{12}, x ) lies in second quadrant.
11
69 If ( alpha ) and ( beta ) are angles in the first quadrant ( tan alpha=frac{1}{7}, sin beta=frac{1}{sqrt{10}} ) then
using the formula ( sin (A+B)= )
( sin A cos B+cos A sin B ) one can find
the value of ( (boldsymbol{alpha}+mathbf{2} boldsymbol{beta}) ) to be
A ( cdot 0^{circ} )
B . ( 45^{circ} )
( c cdot 60 )
D. ( 90^{circ} )
11
70 If ( sin ^{2} theta=frac{1}{4}, ) then the general solution
of ( boldsymbol{theta} )
A ( cdot 2 n pi pm(-1)^{n} frac{pi}{6} )
в. ( frac{n pi}{2} pm(-1)^{n} frac{pi}{6} )
c. ( n pi pm frac{pi}{6} )
D. ( 2 n pi pm frac{pi}{6} )
11
71 Illustration 3.33 Prove that
sin A + sin 2A+ sin 4A + sin 5A
-= tan 3A.
cos A + cos 2A + cos 4A + cos 5A
11
72 If ( sin theta=frac{1}{2}, cos phi=1, ) where ( 0<theta<frac{pi}{2} )
and ( 0<phi leq frac{pi}{2}, ) then ( (cot (theta+2 phi))^{2} ) is
equal to:
11
73 X
3. cose cos/0 – 27 cose + 1
9.
If
then x + y + z is
cosa
2 л
<r), the
COS
O *
equal to
a. 1
b. O
d. none of these
c. – 1
O
11
74 Find the value of ( sin 22^{0} cos 38^{0}+ )
( cos 22^{0} sin 38^{0} )
( A cdot frac{1}{2} )
B. ( -frac{sqrt{3}}{2} )
c. ( frac{sqrt{3}}{2} )
D. ( frac{1}{sqrt{2}} )
11
75 A solution of the equation ( 5 sin ^{2} x+ )
( 3 sin x cos x-3 cos ^{2} x=2 ) is
This question has multiple correct options
A ( cdot_{2 pi+tan ^{-1} frac{-3+sqrt{69}}{6}} )
B. ( quad 7 pi+tan ^{-1} frac{-3-sqrt{69}}{6} )
c. ( tan ^{-1} frac{-3+sqrt{69}}{6}-pi )
D. ( tan ^{-1} frac{-3-sqrt{69}}{6}-5 pi )
11
76 00
5.
For all ein [0, r/ 2] show that, cos (sin o) 2 sin (cos O).
(1981 – 4 Marks)
11
77 If ( sin alpha sin beta-cos alpha cos beta+1=0 ) then
show that ( 1+cot alpha tan beta=0 )
11
78 31. Letf(n) = 2 cos nx ne N, then f (1) f(n + 1)-f(n) is
equal to
Bilma. f(n+3)
b.f(n+2) – O na
c. f(n + 1)f (2) d. f(n + 2)f (2) leto
11
79 if ( sin theta=-frac{4}{5}, pi<theta<frac{3 pi}{2}, ) then find
1. ( sin 2 theta )
2. ( cos 2 theta )
3. ( tan 2 theta )
11
80 The number of value of ( x ) in ( [0,2 pi] )
satisfying the equation
( |cos x-sin x| geq sqrt{2}, ) is
11
81 Solve ( 6 sin ^{2} x+2 sin ^{2} 2 x=5 ) 11
82 (i) If ( sin (theta+alpha)=cos (theta+alpha) ) then
express tan ( theta ) in terms of ( alpha )
(ii) Find the value of ( tan (pi / 4+ )
( theta) cdot tan (pi / 4-theta) )
11
83 sin4t + cost-1
4. The value of 3 –
is equal to
sint + cosºt – 1
11
84 Illustration 3.78
If A + B + C = n, prove that

sin
-sin
= 1 – 2 cos
COS
11
85 Solve: ( cos ^{7}left(sin frac{4 pi}{3}right) ) 11
86 Illustration 2.50 Find the value of x for which f(x) =
sin x – cos x is defined, x € [0,21].
11
87 6. Let n be a positive integer such that sin
+COS
2n
=v”. Then
b. 4 <n<
to gali
a. 6Sn38
c. 45n38
d. 4<n<8 (IIT-JEE 1994)
11
88 71. Which of the following is not the solution of the equation
sin 5x = 16 sinºx (n e Z)?
a. nt
.
b. nt +
c. nt –
d. none of these
11
89 What is the value of ( left(frac{1}{sin 45^{0}}-sin 45^{0}right)left(frac{1}{cos 45^{0}}-cos 45^{0}right. ) 11
90 sin4 x – cos4 x+sin? x cos? x
97. If y=
sin* x + cos* x + sin? scosx*e 0:2), then
a. susc. lsys
c.
VI
ys1
d. none of these
11
91 The value of ( 36^{circ} ) in radians is
A ( cdot frac{pi}{2} )
в. ( frac{2 pi}{5} )
c.
D. ( 3 pi )
11
92 Define radian measure of an angle. 11
93 Prove that ( : frac{sin A}{cot A+operatorname{cossec} A}=2+ )
( frac{sin A}{operatorname{Cot} A-operatorname{cosec} A} )
11
94 94. Ifu=va? cos? + b2 sine + Va? sin? 6 + b2 cos? o,
then the difference between the maximum and minimum
values of u- is given by
a. 2(a² +6²) b. 2 a² +6² .
c. (a + b)2
d. (a – b)?
11
95 Solve ( 1+2 operatorname{cosec} x=-frac{sec ^{2}(x / 2)}{2} ) 11
96 If ( [sin x]+[sqrt{2} cos x]=-3, x in[0,2 pi] )
([.] denotes the greatest integer function), then ( x ) belongs to
A ( cdotleft(pi, frac{5 pi}{4}right) )
B . ( left[pi, frac{5 pi}{4}right] )
c. ( left(frac{5 pi}{4}, 2 piright) )
D・ ( left[frac{5 pi}{4}, 2 piright] )
11
97 15. The respective values of tan A, tan B and tan Care
a. 1, ſ3, ſ b. 1, ſ3, 2
1. c. 1, 2, 13 d. 1, 13, 2 + V3
11
98 9. Prove that 1 + cotes cot – for 0 < €< it. Find when
equality sign holds.
11
99 The value of ( frac{1-tan ^{2} 15}{1+tan ^{2} 15} ) is
( A cdot 1 )
B. ( sqrt{3} )
( c cdot frac{sqrt{3}}{2} )
D.
11
100 46. (1 + tan a tan B)2 + (tan a – tan B)2 =
a. tan’a tan
B b . sec?a secaß
c. tan’a cot+ß iced d. sec’a cos?B
11
101 6. Without using tables, prove that (sin 12°) (sin 48°)
(sin 549) = 1/8.
(IIT-JEE 1980)
11
102 If ( 3 Theta ) is an acute angle, solve the
following equation ( Theta ) :
( (operatorname{cosec} 3 Theta-2)(cot 2 Theta-1)=0 )
11
103 If ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}, boldsymbol{delta} ) are the smallest ( + ) ive angles
in ascending order of magnitude which have their sines equal to a +ive quantity ( lambda ) then the value of ( 4 sin frac{alpha}{2}+ ) ( 3 sin frac{beta}{2}+2 sin frac{gamma}{2}+sin frac{delta}{2}= )
A ( cdot 2 sqrt{1-lambda} )
B. ( 2 sqrt{1+lambda} )
c. ( 2 sqrt{lambda} )
D. ( 2 sqrt{lambda+2} )
11
104 9. sin (90° + 0) is
(a) sin o
(c) -cos e
(b) cos 0
(d) – sin e
11
105 Illustration 2.62 Find the sign of the values of tan 113° –
cos 107° = a and tan 107° – cos 105° = b.
11
106 Solve ( sin ^{4} x+cos ^{4} x=5 / 8 ) 11
107 1. Find the coordinates of the points of intersection of the
curves y = cosx , y= sin 3x if- sus
11
108 Assertion
( (A): frac{tan 3 x-tan 2 x}{1+tan 3 x tan 2 x}=1 Rightarrow x= )
( n pi+frac{pi}{4}, n in l )
Reason
( (R): tan x ) is not defined at ( x=n pi+ )
( frac{pi}{2}, n in l )
A. Both (A) and (R) are individually true and (R) is the correct explanation of (A)
B. Both (A) and (R) are individually true but (R) is not the correct explanation of ( (A) )
C. (A) is true but (R) is false
D. (A) is false but (R) is true
11
109 Find the radian measures
corresponding to the following degree measures:(i) ( 25^{circ}left(text { ii) }-47^{circ} 30^{prime}left(text { iii) } 240^{circ}right.right. )
( (i v) 520^{circ} )
11
110 Prove that
[
frac{sin mathbf{A}+sin 3 A+sin 5 A+sin 7 A}{cos A+cos 3 A+cos 5 A+cos 7 A}=
]
( tan 4 A )
11
111 The value of ( cos left[frac{1}{2} cos ^{-1}left(cos left(frac{-14 pi}{5}right)right)right] ) is/are:
( ^{mathbf{A}} cdot cos left(frac{-7 pi}{5}right) )
B・ ( sin left(frac{pi}{10}right) )
c. ( cos left(frac{2 pi}{5}right) )
D. ( -cos left(frac{3 pi}{5}right) )
11
112 For ( left(boldsymbol{theta}_{1}, boldsymbol{theta}_{2}, boldsymbol{theta}_{3}, ldots ldots boldsymbol{theta}_{n}right) boldsymbol{epsilon}(mathbf{0}, boldsymbol{pi} / 2) ) if
( ln left(sec theta_{1}-tan theta_{1}right)+ln left(sec theta_{2}-right. )
( left.tan theta_{2}right)+ldots . ln left(sec theta-tan theta_{n}right)+ln pi= )
0 then find the value of ( cos left[left(sec theta_{1}+right.right. )
( left.tan theta_{1}right)left(sec theta_{2}+tan theta_{2}right) dots . .left(sec theta_{n}+right. )
( left.tan theta_{n}right) )
11
113 If the radian measures of two angles of a triangle are as given below. Find the radian measure and the degree measure of the third angle.
(i) ( frac{5 pi}{9}, frac{5 pi}{18} )
(ii) ( frac{3 pi}{5}, frac{4 pi}{15} )
11
114 14. Prove that 2 cos 2″ @ +1
— = (2cos 6 – 1)(2cos 20 – 1)
2 cos 0+1
2012
(cos 22 0 – 1) … (2cos 2n-1 0-1).
11
115 Determine range ( boldsymbol{y}=mathbf{3} sin boldsymbol{x}+ )
( 4 cos (x+pi / 3)+7 )
A ( cdot 7-sqrt{5-3 sqrt{3}}, 7+sqrt{5-3 sqrt{3}} )
B・ ( 7-sqrt{10-3 sqrt{3}}, 7+sqrt{10-3 sqrt{3}} )
c. ( _{7}-sqrt{10+3 sqrt{3}}, 7+sqrt{10-3 sqrt{3}} )
D ( cdot 7-sqrt{10+3 sqrt{3}}, 7-sqrt{10-3 sqrt{3}} )
11
116 =an
tany
, (
By) then sin 2a+ sin2B+
tan ß
tan (a + ß – y)
tan (a – B + 7)
sin2y=
a. 0
c. 2
b. 1 o lepszywe
d. 1/2
11
117 Illustration 3.72 Prove that sin 0 + sin 30 + sin 50 + …
sinʼne
+ sin(2n-1) 0=
sin e
11
118 find the value
( tan 11 frac{pi}{3} )
11
119 ( sin 3 x+sin x+2 cos x=sin 2 x+ )
( 2 cos ^{2} x )
11
120 71. iftam B-2 sin a sin ycose( a + y, then coto, coth, coty
71. If tan ß= 2 sin a sin y cosec(a + ), then cot a, cot B, coty
o are in
a. A.P.
b. G.P.
c. H.P.
d. none of these
11
121 The measure of an angle in degrees, grades and radians be ( mathrm{D}, mathrm{G} ) and ( mathrm{C} )
respectively, then relation between them ( frac{boldsymbol{D}}{mathbf{9 0}}=frac{boldsymbol{G}}{mathbf{1 0 0}}=frac{boldsymbol{2 C}}{boldsymbol{pi}} ) but ( mathbf{1}^{circ}= )
( left(frac{180}{pi}right)^{0} simeq 57^{circ}, 17^{prime}, 44.8^{prime prime} ) and sum of
interior angles of a ( n ) -sided regular polygon is ( (2 n-4) frac{pi}{2} . ) On the basis of above information, answer the following questions :The number of sides of two
regular polygon are as 5: 4 and the difference between their angles is ( frac{pi}{20} )
then the number of sides in the
polygons respectively are –
A. 25, 20
B. 20, 16
c. 15,12
D. 10,8
11
122 The general value of ( x ) for the equation
( 9^{cos x}-2.3^{cos x}+1=0 )
A ( . n pi )
в. ( frac{n pi}{2} )
( c cdot 2 n pi )
D. ( (2 n+1) frac{pi}{2} )
11
123 Illustration 3.35 Prove that (cos a + cos 3)2 + (sin a + sin B)?
= 4 cos? (a-B)
11
124 а
76. If 0=3a and sin
=-
=, the value of the expression
x 22 the
a cosec a – b sec a is
a
b. 2ſa² +6²
c. a + b
d. none of these
11
125 8. If a < 3 cos x + 5 sin(x – 1/6) < b for all x, then (a, b) is
a. (-V19, 719) b. (-17, 17)
c. (-√21, √21) d. none of these
11
126 29. The system of equations tan x = a cot x, tan 2x = b cos y
a. cannot have a solution if a=0
b. cannot have a solution if a=1
c. cannot have a solution if 2Va > b(1 – a)
d. has a solution for all a and b
11
127 Change the following radian measure to degree measure:
( frac{3 pi}{2} )
A ( cdot 120^{circ} )
( ^{circ} )
B ( .240^{circ} )
( c .270^{circ} )
D. ( 300^{circ} )
11
128 Convert ( left(frac{5 pi}{6}right)^{c} ) into degrees. 11
129 Prove that:
( 2 sin ^{2} frac{pi}{6}+operatorname{cosec}^{2} frac{7 pi}{6} cos ^{2} frac{pi}{3}=frac{3}{2} )
11
130 ( frac{sec 8 theta-1}{sec 4 theta-1}=frac{tan r theta}{tan 2 theta} cdot ) Find ( r ) 11
131 2 TC
210
Illustration 3.93
Prove that 4cos
.cos–1=2 cos-
11
132 27. The greatest value of sin+e+ cose is
a. 1/2
b. 1
c. 2
d. 3
11
133 Illustration 3.83 If cos (A + B+C)=cos A cos B cos C, then
8 sin (B + C) sin (C + A) sin (A + B)
find the value of –
sin 2 A sin 2B sin 2C
11
134 Solve the following equation:
( cos x=frac{1}{2} )
11
135 14. If sin 0+ cos 0 = – and 0 s < , then tan o is
a. – 4/3
b. – 3/4
c. 3/4
d. 4/3
11
136 67. If o is an acute angle and
tan 0 + cot 0 = 2, then the value
of tans 0 + cot5 O is
(1) 1
(2) 2
(3) 3
(4) 4
11
137 ( tan ^{-1}left[frac{a cos x-b sin x}{b cos x+a sin x}right]=tan ^{-1}left(frac{a}{b}right) )
( mathcal{L} )
A. True
B. False
11
138 If ( cos left(65^{0}-Aright) cos left(25^{0}+Bright)- )
( sin left(65^{0}-Aright) sin left(25^{0}+Bright)=sin (m+ )
( A-B) ).Find ( m )
11
139 Given ( frac{pi}{2}<alpha<pi, ) then the expression ( sqrt{frac{1-sin alpha}{1+sin alpha}}+sqrt{frac{1+sin alpha}{1-sin alpha}}= )
A ( frac{1}{cos alpha} )
B. ( -frac{2}{cos alpha} )
c. ( frac{2}{cos alpha} )
D. None of these
11
140 Express the following angles in degrees.
( begin{array}{ll}text { (1) } & left(frac{5 pi}{12}right)^{circ}end{array} )
(2) ( -left(frac{7 pi}{12}right)^{circ} )
(3) ( frac{pi}{3} )
(4) ( frac{5 pi^{circ}}{6} )
(5) ( frac{2 pi^{circ}}{9} )
(6) ( frac{7 pi^{circ}}{24} )
11
141 Solve ( sin x+sqrt{3} cos x geq 1 ) 11
142 90. If cosA + cos²B + cos²C = 1, then A ABC is
a. equilateral b . isosceles
c. right angled d. none of these
11
143 Illustration 3.21 Prove that (1 + tan 1°)(1 + tan 2°) …
(1 + tan 45º = 223.
11
144 8. Prove that the equation 2 sin x = (x + a has no solution for
ae
3
Donne
o
ttomu
11
145 ( sin 11^{circ} 19 cos 18^{circ} 41+ )
( cos 11^{circ} 19 sin 18^{circ} 41= )
A .
B. ( frac{sqrt{3}}{2} )
( c cdot frac{1}{2} )
D. 0
11
146 ( boldsymbol{A}+boldsymbol{B}=frac{boldsymbol{pi}}{mathbf{3}} ; cos boldsymbol{A}+cos boldsymbol{B}=mathbf{1}, ) value of
( |cos A-cos B| ) is
A ( cdot frac{1}{3} )
B. ( sqrt{frac{2}{3}} )
( c cdot sqrt{frac{3}{2}} )
D.
11
147 For an acute angle, ( alpha, sin alpha+cos alpha )
takes the
greatest value when ( alpha ) is
A ( .30^{circ} )
B . ( 45^{circ} )
( c cdot 60^{0} )
D. ( 90^{circ} )
11
148 ( tan 100^{circ}+tan 125^{circ}+ )
( tan 100^{circ} tan 125^{circ} ) is equal to
A . 0
B. ( frac{1}{2} )
( c .-1 )
D.
11
149 Illustration 3.73
Prove that
cos 3x
sin 2x sin 4x
cos 5x
sin 4x sin 6x
cos 7x
+ –
sin 6x sin 8x
+
cos 9x
sin 8x sin 10x
– (cosec x) [cosec 2x – cosec 10x]
11
150 Illustration 4.3 Solve tan x + tan 2x + tan 3x = tan x tan 2x
tan 3x, xe [0, 1].
11
151 If ( 0 leq x leq pi ) and ( 81^{sin ^{2} x}+81^{cos ^{2} x}=30 )
then ( x ) is equal to
This question has multiple correct options
A ( cdot frac{pi}{6} )
в.
c. ( frac{5 pi}{6} )
D. ( frac{2 pi}{3} )
11
152 +
54. Tet (1-сos 2x+sin 2x (1+cotx + cot’ x
(1+cos 2x + sin 2x ) ( 1+tan x + tan- x
then the minimum value of P(x) equals sro
a. 1
b. 2
c. 4
d. 16
11
153 27. If COS X _ cos(x+0)_cos(x +20)
cos(x +30)
d
then
a
b
ngh atc :
is equal to
b+d
b. –
11
154 5. Let f:(-1, 1) + R be such that f(cos 40) = 5
2-sece
(
π
π
for
for 0 €(0.4)-(* 1). Then the v
e
1. Then the value(s) of
is (are)
b. 1+
c.
1-
d. 1+
11
155 Which of the following is least? (All angles have been measured in radians)
( A cdot sin 3 )
B. ( sin 2 )
( c cdot sin 1 )
D. ( sin 7 )
11
156 satisfying
4. Find all values of 0 in the interval
the equation
(1 – tan 0) (1 + tan ) sec+ 2 tan’e = 0.
11
157 If ( theta_{1}, theta_{2}, theta_{3}, ldots . theta_{n} ) are in ( A . P ., ) then
( frac{sin theta_{1}+sin _{2}+ldots+sin theta_{n}}{cos theta_{1}+cos theta_{2}+.+cos theta_{n}}= )
( A cdot 0 )
B ( cdot tan left(theta_{1}+theta_{n}right) )
c. ( tan left(frac{theta_{1}+theta_{n}}{2}right) )
D. ( tan left(frac{theta_{n}-theta_{1}}{2}right) )
11
158 If ( sin x=sin y ) and ( cos x=cos y, ) then
( x ) is
A ( .2 n pi+y )
в. ( 2 n pi-y )
c. ( n pi+y )
D. ( n pi-y )
11
159 Given that ( cos 50^{circ} 20^{prime}=0.6388 ) and
( cos 50^{circ} 42^{prime}=0.6334, ) then the possible
value of ( cos 50^{circ} 20^{prime} ) is
A .0 .6293
B. 0.6307
c. 0.6361
D. 0.6414
11
160 Find the general solution of the
equation ( 4 cos ^{2} x=1 )
11
161 OS
у
4. If sinx + cosx = y + – for x € [0, 1], then
a. x = 7/4
b. y = 0
c. y = 1
d. x = 370/4
11
162 18. If (1 + sin t) (1 + cot t) =
– then find the value of
(1-sin t) (1 – cos t).
11
163 32. If 0< 0< t, then minimum value of 3 sin 0+ cosec O is
a. 4
b. 3
c. 5
d. 6
11
164 88. In triangle ABC,
csin A+sin B + sin C
sin A+ sin B-sin C
is equal to
a. tan
cot”
2
2
А
В
cot cota
d. tantan
11
165 Find the value of ( tan 9^{circ}-tan 27^{circ}- )
( tan 63^{circ}+tan 81^{circ} )
( mathbf{A} cdot mathbf{0} )
B . 2
c. 1
D. 4
11
166 50. One of the general solutions of 4 sinºx + cos x = 1 is
a. nt = a/2, a=cos (1/5), V ne z
b. nn a/2, a= cos(3/5), V ne z
c. 2n = a/2, a= cos'(1/3), V ne z
d. none of these
11
167 Solve the following equation:
( cos x=sqrt{3} )
11
168 ( f sin left(60^{circ}+30^{circ}right)=sin 60^{circ} cos 30^{circ}+ )
( sin 30^{circ} cos 60^{circ} ) then what is the value of
( sin left(60^{circ}+30^{circ}right) .=? )
11
169 Illustration 3.8 In AABC, if cot A + cot B + cot C = 0 then
find the value of cos A cos B cos C. 20
11
170 Prove that
( cos frac{pi}{15} cos frac{2 pi}{15} cos frac{3 pi}{15} cos frac{4 pi}{15} cos frac{5 pi}{15} cos frac{6}{1} )
1
( 2^{7} )
11
171 Illustration 3.25
Find the range of
f(x)=
(cos x – 3)2 + (sin x +4)2
11
172 1. (1+cos 1 + cos y 1 + cos 4 + cos?”) is equal
– COS
(1984 – 3 Marks)
(6) cos.
(d) 1+ v2
(d) 22
11
173 The number of solutions of the equation ( 2 cos left(frac{x}{2}right)=5^{x}+5^{-x} ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. None of these.
11
174 3-tan?
55. If –
= k cos – then the value of k is
1 – tan2
a. 1
c. 3
b. 2
op d. 4
11
175 ( cos ^{2} 42^{circ}-sin ^{2} 48^{circ} ) 11
176 Illustration 3.91
1/16.
Prove that sin 6° sin 42° sin 66° sin 78°
11
177 21. Which of the following is not the value of sin 27° –
cos 27°?
b. – V5-15
2
c.- 212
d. none of these
11
178 74. The smallest positive x satisfying the equation
logcosxsin x + logsinr cos x = 2 is
a. Td/2
b. /3
c. 77/4 shot
d. Td/6
11
179 Illustration 3.49 Find the maximum and minimum values
of cos 0 – 6 sin cos 0+3 sin’e+2.
11
180 4. If 3 tan A + 4 = 0, then the value of 2 cot A-5 cos A+
sin A is equal to
4. If3 tan A + 4 = 0, then the value of 2 cot A -5 cos A+
a.
if –
<A<TT
<A<2a
2
<A< T
d.
<A< 211
10
10
11
181 14.
The general values of
2sin20-3sino-2=0 is
satisfying the equation
(1995)
(b) na +(-1)”</2
(a) na +(-1)” x/6
©
nt+(-1)"51/6
(d) nt+(-1)" 770/6
11
182 The degree measure of 1 radian (taking ( left.boldsymbol{pi}=frac{boldsymbol{2} boldsymbol{2}}{boldsymbol{7}}right) ) is
A ( cdot 55^{circ} 61^{prime} 22^{prime prime} ) (approx.)
В ( cdot 57^{circ} 16^{prime} 22^{prime prime} ) (approx.)
c. ( 57^{circ} 22^{prime} 16^{text {” }} ) (approx.)
D. ( 57^{circ} 22^{prime} 22^{prime prime} ) (approx.)
11
183 If ( 2 tan beta+cot beta=tan alpha, ) prove that
( cot beta=2 tan (alpha-beta) )
11
184 If ( 0<x leq frac{pi}{2}, ) then ( (sin x+operatorname{cosec} x) ) is
greater than or equal to
A . 0
B.
( c cdot 2 )
D. None of these
11
185 If ( x sin ^{3} theta+y cos ^{3} theta=cos theta sin theta ) and
( boldsymbol{x} sin boldsymbol{theta}=boldsymbol{y} cos boldsymbol{theta} operatorname{then} boldsymbol{x}^{2}+boldsymbol{y}^{2}=mathbf{1} )
If the statement is True, enter 1 , else
enter 0
A.
11
186 If ( sin ^{2} theta+5 cos ^{2} theta=4, ) then find ( theta ) and
hence prove that ( sec theta+operatorname{cosec} theta=2+ )
( frac{2}{sqrt{3}} )
11
187 13.
tan x whe
Show that the value of a -, wherever defined never lies
tan 3x
between – and 3.
(1992 – 4 Marks)
11
188 If ( tan x=-frac{3}{4}, frac{3 pi}{2}<x<2 pi, ) then find
( cos 2 x )
11
189 Prove that:
( sin 50^{circ}+sin 10^{circ}=cos 20^{circ} )
11
190 66. Which one of the following is true
for 0° < cos20
(3) cose cos20
11
191 As ( theta ) increases from ( frac{pi}{4} ) to ( frac{5 pi}{4}, ) the value
of ( 4 cos frac{1}{2} theta )
A. increases, then decreases
B. decreases, and then increases
C. decreases throughout
D. increases throughout
E. decreases, increases, and then decreases again
11
192 3. The general value of 0 satisfying the equation tan²0 +
sec 20= 1 is
(IIT-JEE 1996)
11
193 cos x
18. If
sin x
siny
1
2
3
= =, where x, y e
2′
1, then the
cos y
value of tan(x + y) is equal to
a. 113
c. 17
b. 114
d. 115
11
194 46._
sin? A-sin-B
is equal to
sin Acos A-sin B cos B
a. tan(A – B)
c. cot(A – B)
= b. tan(A + B)
d. cot(A + B)
11
195 13. 3 (sin x -cos x)* + 6 (sin x + cos x)2 + 4 (sinºx+cos© x) =
(1995)
(a) 11 (6) 12 C 13 (d) 14
11
196 ff ( boldsymbol{x}=boldsymbol{a} sin boldsymbol{theta}+boldsymbol{c} cos boldsymbol{theta} ) and ( boldsymbol{y}= )
( boldsymbol{a} cos boldsymbol{theta}-boldsymbol{c} sin boldsymbol{theta}, operatorname{then} boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{a}^{2}+boldsymbol{c}^{2} )
A . True
B. False
11
197 In the fourth quadrant, and
trigonometric ratios are positive
A . ( cos ), sec
B. sin, cos
( c cdot sin , operatorname{cosec} )
D. tan, cot
11
198 ( frac{sin (n+1) A+2 sin n A+sin (n-1) A}{cos (n-1) A-cos (n+1) A} )
is equal to
A ( cdot tan frac{A}{2} )
B. ( cot frac{A}{2} )
( c cdot tan A )
D. ( cot A )
11
199 If ( A ) and ( B ) are acute positive angles satisfying the equations ( 3 sin ^{2} A+ ) ( 2 sin ^{2} B=1 ) and ( 3 sin 2 A-2 sin 2 B= )
( 0, ) then ( A+2 B ) is equal to
A ( cdot frac{pi}{4} )
в.
( mathrm{c} cdot_{3} frac{pi}{4} )
D. ( frac{2 pi}{3} )
11
200 then find the range of
Illustration 3.4 If sin a cos B=-
values of cos a sin ß.
11
201 If ( x=sin 1, y=sin 2 ; z=sin 3 ) then
A. ( x<yy>z )
c. ( y<z<x )
D. ( z<x<y )
11
202 The value of
( sin 10^{circ} sin 30^{circ} sin 50^{circ} sin 70^{circ} ) is:
( ^{A} cdot frac{1}{36} )
в. ( frac{1}{32} )
c. ( frac{1}{18} )
D. ( frac{1}{16} )
11
203 31. Number of real solutions of the equation (tan x + 1)
(tan x + 3) (tan x + 5) (tan x + 7) = 33
a. will be two in the interval [– 1/2, 7/2]
b. will be four in the interval [- r/2, 1/2]
c. will be three in the interval (-1/2, 1)
d. will be four in the interval (-1/2, 1)
11
204 show that ( sqrt{2+sqrt{2+sqrt{2+2 cos 8 theta}}}= )
( 2 cos theta, 0<theta<frac{pi}{8} )
11
205 Illustration 3.83 If cos (A+B+C)=cos A cos B cos C, then
8 sin (B + C) sin (C + A) sin (A + B)
find the value of
sin 2 A sin 2B sin 2C
11
206 Illustration 3.79 In any triangle ABC, prove that
sin’A cos(B – C) + sin’B cos(C – A) + sinC cos(A – B)
= 3 sin A sin B sin C
11
207 State true or false.
If ( left(1+sin ^{2} thetaright)=3 sin theta cos theta ) then
( tan theta=-1 o r frac{1}{2} )
A . True
B. False
11
208 Prove the following Identities ( frac{tan alpha+tan beta}{cot alpha+cot beta}=tan alpha tan beta ) 11
209 Solve the following equation:
( sin x=frac{sqrt{2}}{2} )
11
210 Illustration 4.36 Find common roots of the equations 2sin²x
+ sinº 2x = 2 and sin 2x + cos 2x = tan x.
11
211 2. The value of tan ß is
sin a(1+ Acos B)
Acos a cos ß
sin a(l – Acos B)
Acos a cos ß
cosa(1+ Asin ß)
A cos a cos ß
cosa(1 – Asin B)
Acos a cos ß
11
212 If ( boldsymbol{alpha} boldsymbol{epsilon}left(boldsymbol{0}, frac{boldsymbol{pi}}{2}right), ) then the expression
( sqrt{x^{2}+x}+frac{tan ^{2} x}{sqrt{x^{2}+x}} ) is always greater
than or equal to
( A cdot 2 tan alpha )
B. 2
( c . )
( D cdot sec ^{2} alpha )
11
213 In which quadrant does the terminal
side of the angle ( 250^{0} ) lie?
A. Quadrant III
B. Quadrant I
c. Quadrant
D. Quadrant IV
11
214 Illustration 2.44
Find the range of f(x) =
4 cos x
3
11
215 What is ( (sin x cos y+ )
( cos x sin y)(sin x cos y-cos x sin y) )
equal to?
A ( cdot cos ^{2} x-cos ^{2} y )
B. ( cos ^{2} x-sin ^{2} y )
( c cdot sin ^{2} x-cos ^{2} y )
D. ( sin ^{2} x-sin ^{2} y )
11
216 Let ( alpha ) and ( beta ) be any two positive values
of ( x ) for which ( 2 cos x,|cos x|, ) and ( 1- )
( 3 cos ^{2} x ) are in G.P. The minimum value
of ( |boldsymbol{alpha}-boldsymbol{beta}| ) is
A ( cdot frac{pi}{3} )
в.
c. ( frac{pi}{2} )
D. None of these
11
217 24. If a = *then the value of (tan a tan 2c + tan 2a tan 4a.
14°
+ tan 4a tan a) is
a. 1
b. 1/2
c. 2
d. 1/3
11
218 Find all pairs of ( x, y ) that satisfy the
equation ( cos x+cos y+cos (x+y)=-3 / 2 )
11
219 The value of ( sin ^{2} 30^{circ}-cos ^{2} 30^{circ} ) is:
A. ( -frac{1}{2} )
B. ( frac{sqrt{3}}{2} )
( c cdot frac{3}{2} )
D. ( frac{2}{3} )
11
220 State whether the following statement is true or false.
( frac{cos A-sin A+1}{cos A+sin A-1}=operatorname{cosec} A+cot A )
(by using the identity ( operatorname{cosec}^{2} boldsymbol{A}=1+ )
( left.cot ^{2} A .right) )
A . True
B. False
11
221 Illustration 3.69 Show that
4 sin 27° = (5+15)1/2 – (3-55)12.
11
222 If ( boldsymbol{x}=boldsymbol{a}+boldsymbol{b} boldsymbol{omega}+boldsymbol{c} boldsymbol{omega}^{2}, boldsymbol{y}=boldsymbol{a} boldsymbol{omega}+boldsymbol{b} boldsymbol{omega}^{2}+boldsymbol{c} )
and ( z=a omega^{2}+b+c omega ) then find the
value of ( frac{x^{2}}{y z}+frac{y^{2}}{z x}+frac{z^{2}}{x y} )
11
223 If ( frac{cos (theta-alpha)}{sin (theta+alpha)}=frac{m+1}{m-1}, ) then ( m ) is equal
to
A ( cdot tan left(frac{pi}{4}-thetaright) tan left(frac{pi}{4}-alpharight) )
B. ( tan left(frac{pi}{4}-thetaright) tan left(frac{pi}{4}+alpharight) )
c. ( tan left(frac{pi}{4}+thetaright) tan left(frac{pi}{4}+alpharight) )
D. ( tan left(frac{pi}{4}+thetaright) tan left(frac{pi}{4}-alpharight) )
11
224 If ( sin x+sin ^{2} x=1, ) then ( cos ^{2} x+cos ^{4} x )
is :
( A )
B. 2
( c cdot 3 )
D. 4
11
225 Illustration 3.61 Evaluate cos a cos 2a cos 3a … cos 999a,
211
where a=
1999
11
226 ( cos ^{4} theta-sin ^{4} theta+1 ) is equal to:
A ( cdot 2 cos e c^{2} theta )
B. ( -2 cos e c^{2} theta )
c. ( frac{2}{tan ^{2} theta} )
D. ( frac{2 cot ^{2} theta}{cos e c^{2} theta} )
11
227 4. In which of the following sets the inequality sinºx + cosºx
> 5/8 holds good?
a. (-1/8, 7/8)
b. (37/8, 57/8)
c. (Tt/4, 31/4)
d. (771/8, 9/8) .
11
228 19. Show that
1+ sin A
cos A 1

+-
cos B
-sin B
2 sin A-2 sin B
sin(A-B)+cos A – cos B
11
229 2. Which of the inollowing is collectin 1° sin 1
b. sin 1° < sin 1
c. sin 1° = sin 1
b
d. sin 1° =
*
sin 1
180
11
230 All the angles between and which
satisfy ( 90^{circ} ) the ( 0^{circ} ) equation ( sec ^{2} theta cdot cos e c^{2} theta+2 cos e c^{2} theta=8 )
11
231 If ( sec A=operatorname{cosec} B=frac{5}{3}, ) then the value
of ( (boldsymbol{A}+boldsymbol{B}) ) is equal to
( mathbf{A} cdot mathbf{0} )
B. ( 90^{circ} )
( mathbf{c} cdot90^{circ} )
11
232 85. If x = r cos O cos 0, y = r cos
sin 0 and 2 = r sin 0, then the
value of x2 + y + z is
(1) 2
(2)
11
233 equals
38. The value of expression
2(sin 1° + sin 2° + sin 3° + … + sin 89°)
2
2(cos 1° +cos 2° +…+cos 44°)+1
a. 12
b. 1712
c. 1/2
d. O
10
11
234 If ( sin x-cos y=sin frac{pi}{7} ) and ( cos x+ )
( sin y=cos frac{pi}{7} ) then find the value of
( sin (x-y) )
11
235 If ( sin A=frac{5}{13} ) then evaluate ( cos A ) and
( tan A )
11
236 ( frac{sin A}{sin left(90^{circ}-Aright)}+frac{cos A}{cos left(90^{circ}-Aright)}= ) 11
237 Illustration 4.22 Find the general values ofx and y satisfying
the equations 5 sin x cos y = 1 and 4 tan x = tany.
11
238 If ( tan (A+B)=p, tan (A-B)=q )
then prove that ( tan 2 A=frac{p+q}{1-p q} )
11
239 Illustration 4.39
Solve 7 cos²0 + 3 sin²0= 4
11
240 54. The value of
(+tan? 2°. tan88°
(1) 1
(2) 2
(3) O
(4) 4
11
241 Show that ( frac{2 tan 30^{circ}}{1-tan ^{2} 30^{circ}}=sqrt{3} ) 11
242 13. If O E TO, 57) and r e R such that 2sin e = r4 – 2r-
+ 3 then the maximum number of values of the pair
(r, ) is
11
243 Solve ( : log _{frac{x^{2}-6 x}{10}}(sin 3 x+sin x)= )
( log _{frac{x^{2}-6 x}{10}}(sin 2 x) )
A. ( _{x=frac{5 pi}{3}} )
в. ( x=-frac{5 pi}{3} )
c. ( x=-frac{2 pi}{3} )
D. ( x=-frac{4 pi}{3} )
11
244 Prove ( frac{cos 20^{circ}+sin 20^{circ}}{cos 20^{circ}-sin 20^{circ}}=tan 65^{circ} ) 11
245 Find ( cos ^{4}(pi / 8)+cos ^{4}(3 pi / 8)+ )
( cos ^{4}(5 pi / 8)+cos ^{4}(7 pi / 8)-3 / 2 )
11
246 Solve the equation ( sqrt{3} cos x-sin x=1 )
A ( cdot x=2 n pi-frac{pi}{3} pm frac{pi}{3}, n in Z )
В . ( x=2 n pi-frac{pi}{6} pm frac{pi}{3}, n in Z )
c. ( x=2 n pi-frac{pi}{6} pm frac{pi}{6}, n in Z )
D. None of these
11
247 Solve: ( 2(sin x-cos 2 x)-sin 2 x(1+ )
( 2 sin x)+2 cos x=0 )
11
248 solve:-
( 1+cos 2 x+cos 4 x+cos 6 x= )
A ( .2 cos x cos 2 x cos 3 x )
B. 2 ( cos x sin 2 x cos 3 x )
( mathrm{c} cdot 4 cos x cos 2 x cos 3 x )
D. ( 4 cos x sin 2 x sin 3 x )
11
249 Which of the following is correct?
( A cdot sin 1^{circ}>sin 1 )
B. ( sin 1^{circ}<sin 1 )
( mathbf{c} cdot sin 1^{circ}=sin 1 )
D ( cdot sin 1^{circ}=frac{pi}{180} sin 1 )
11
250 5. The number of solutions of the equation tan x + sec x =
2 cos x lying in the interval [0, 21) is
a. O
b. 1
c. 2
d. 3 (IIT-JEE 1993)
11
251 4. Given a + B – y = 1, prove that sin’a + sin?B – sin?y=
2 sin a sin ß cos y.
(IIT-JEE 1980)
11
252 6.
Without using tables, prove that
(1982 – 2 Marks)
(sin 12°) (sin 48) (sin
11
253 36. The general solution of tan 0 + tan 20 + tan 30= 0 is
a. 0=nt/6, ne Z only
b. O=nt &, ne Z, where tan a= 1/
c. Both a and b
d. none of these
11
254 Convert ( 290^{circ} ) into radian measure 11
255 Illustration 2.29 Suppose the point with coordinates (-12,5)
is on the terminal side of angle 8. Find the values of the six
trigonometric functions of e.
11
256 ( operatorname{can} tan 65^{0}=tan 25^{0}+tan 40^{0} ? )
If Yes answer is ( 1, ) else 0
11
257 Illustration 3.95
Illustration 3.95
* 5 = 1, then find the range of 2x + y.
=1, then find the range of 2x + y.
11
258 Express the following angles in radian
measure:
i) ( 520^{circ} )
ii) ( -310^{circ} )
iii) ( 630^{circ} )
iv) ( -22^{circ} 30^{prime} )
11
259 If ( sin A=frac{1}{2} ) and ( cos B=frac{sqrt{3}}{2} ) where ( A )
lies in second quadrant and B lies in first quadrant, find the values of (a) ( sin (A-B)(b) cos (A-B)(c) tan (A+ )
( boldsymbol{B}) )
11
260 50. If tan Atan B = =, then (5 – 3 cos 2A) (5 – 3 cos 2B) =
a. 2
b. 8
d. 16
11
261 If ( tan ^{2} ) la ( tan ^{2} beta+tan ^{2} beta cdot tan ^{2} gamma+ )
( tan ^{2} gamma cdot tan ^{2} alpha+2 tan ^{2} alpha cdot tan ^{2} beta cdot tan ^{2} gamma= )
1, then the value of ( sin ^{2} alpha+sin ^{2} beta+ )
( sin ^{2} gamma ) is :
A .
B. –
( c )
( D cdot frac{1}{2} )
11
262 1.
The period of sine is
(a) a? (b) a
©
20
[2002]
(d) a 12
11
263 Given ( sin phi=frac{15}{17}, ) find the value of ( frac{3-4 sin ^{2} phi}{4 cos ^{2} phi-3} ) 11
264 If ( 2 sin left(theta+frac{pi}{3}right)=cos left(theta-frac{pi}{6}right), ) prove
that ( tan theta+sqrt{3}=0 )
11
265 Illustration 4.45
Solve 13 cos 0 – 3 sin 0 = 4 sin 20 cos 30.
11
266 In the figure given above, ( A B ) is paralle
to ( C D . ) If ( angle D C E=x ) and ( angle A B E=y )
then what is ( angle C E B ) equal to?
A ( cdot y-x )
в. ( frac{(x+y)}{2} )
c. ( x+y-(pi / 2) )
D. ( x+y-pi )
11
267 41. If cos? x -(c – 1) cos x + 2c 26 for every x e R, then the
true set of values of c is
a. [2,-)
D. 14, ) GHT 02
c. (-∞, -21
d. (-0, 4]
11
268 17. If и, = sin”Ө + cos”Ө, thеn рrоvе thаt 5
и; – и, — и
1 = 3.
и — и, и
11
269 79. If V2 cosA=cosB + cos’ B, and V2 sin A=sin B-sin’ B
then sin (A – B) =
a. +1
bゃう
c. +
atd.
d.t-
11
270 Simplify: ( sin ^{-1}(cos x) ) 11
271 ( operatorname{In} ) any triangle ( prodleft(frac{sin ^{2} B+sin B+1}{sin B}right) )
is always greater than
( A cdot 9 )
B. 3
c. 27
D. None of these
11
272 Prove ( : sqrt{frac{1+sin A}{1-sin A}}=sec A+tan A ) 11
273 If ( tan theta+frac{1}{tan theta}=2, ) find the value of
( cot ^{2} theta+frac{1}{cot ^{2} theta} )
11
274 5. Let 0 < x < 7:/4, then (sec 2x -tan 2x) equals
X
in
x
+
11
275 ( sin ^{-1} frac{2 a}{1+a^{2}}+cos ^{-1} frac{1-b^{2}}{1+b^{2}}= )
( 2 tan ^{-1} x )
11
276 Prove that ( frac{cos 9^{0}+sin 9^{0}}{cos 9^{0}-sin 9^{0}}=cot 36^{0} ) 11
277 alue for which tan o– 1, cos o-ta
2. The most general value for which tan O=-1, cos O=
is (n e Z)
a. nt+
b. nt + (-1)” ITT
c. 2n1 + 711
d. none of these
11
278 14. The value of tan A tan B + tan B tan C + tan C tan A IS
a. 5 – 4 13 b. 5 +4 13
c. 6 + √3
d. 6-√3
11
279 10
2. Compute tan 22
11
280 2. Given to the
3.
Given a+B-y=t, prove that
sina + sin B-sin?y=2 sina sin
prove singsing cosy
cosy
(1980)
1980)
11
281 tan 20
15. Prove that an 2 = (1 + sec 20) (1 + sec 220)
tan e
(1 + sec 230) … (1 + sec 2″0).
11
282 If ( 0 leq x leq 2 pi, ) then the number of
solutions of the equation ( sin ^{6} x+ )
( cos ^{6} x=1 ) is
( A cdot 2 )
B. 3
( c cdot 4 )
D. 5
E . 8
11
283 Three roots of the equation, ( x^{4}-p x^{3}+ )
( boldsymbol{q} boldsymbol{x}^{2}-boldsymbol{r} boldsymbol{x}+boldsymbol{s}=boldsymbol{0} ) are ( tan boldsymbol{A}, tan boldsymbol{B} boldsymbol{&} )
( tan C ) where ( A, B, C ) are the angles of ( a ) triangle. The fourth root of the bi
quadratic is
A ( frac{s^{2}-s q+s}{r+(s-q) p} )
B. ( frac{s^{2}+s q+s}{r+(s+p) q} )
c. ( frac{s^{2}-s q+s}{r-(s-p) q} )
D. ( frac{s^{2}-s q-s}{r-(s-q) p} )
11
284 Sin
56. The total number of solutions of cotx| = cotx +
e [0, 31), is equal to
a. 1
b. 2
c. 3
d. 0
11
285 If lies in the second quadrant, then the value of ( sqrt{left(frac{1-sin theta}{1+sin theta}right)}+sqrt{left(frac{1+sin theta}{1-sin theta}right)} ) is
( mathbf{A} cdot 2 sec theta )
в. ( -2 sec theta )
( c .2 operatorname{cosec} theta )
D. noneofthese
11
286 13. One root of the equation cox – * * } – oles in the
73. One root of the equation cos x – x + – = 0 lies in the
interval
2
T,
11
287 m=0
1. Suppose sin’x sin 3x = Cocos mx is an identity in x,
where Co, C, … ,C, are constants, and Cn #0, then the
value of n is
(IIT-JEE 1981)
11
288 7. Solve the equation 2 sinx + cos y = 2 for the values of x
and y.
11
289 Is it right to say that ( sin (A+B)= )
( sin A+sin A . ) Justify your answer?
11
290 25. If x and y are positive acute angles such that (x + y) and
(x – y) satisfy the equation tan-0 – 4 tan 0+1= 0, then
b. y =
d. x
11
291 If ( cos x=tan y, cot y=tan z ) and
( cot z=tan x ; ) then ( sin x= )
A. ( frac{sqrt{5}+1}{4} )
B. ( frac{sqrt{5}-1}{4} )
c. ( frac{sqrt{5}+1}{2} )
D. ( frac{sqrt{5}-1}{2} )
11
292 If ( sin x+sin y=sqrt{3}(cos y-cos x) )
then value of ( sin 3 x+sin 3 y ) equals
( A cdot 3 )
B. ( sqrt{3} )
( c cdot 0 )
D.
11
293 Prove that ( tan (x-y)=frac{tan x-tan y}{1+tan x tan y} ) 11
294 ff ( boldsymbol{y}=sec ^{2} boldsymbol{Theta}+cos ^{2} boldsymbol{Theta}, boldsymbol{0} neq boldsymbol{0}, ) then
A. ( y=0 )
B. ( y leq 0 )
( mathbf{c} cdot y>2 )
D. ( y neq 2 )
11
295 The range of ( f(x)=cos ^{2} x+sec ^{2} x ) is
( [a, infty] ) Find a
11
296 If ( tan A-tan B=x ) and ( cot B- )
( cot A=y ) then ( cot (A-B)= )
A ( cdot frac{1}{y}-frac{1}{x} )
в. ( frac{1}{x}-frac{1}{y} )
c. ( frac{1}{x}+frac{1}{y} )
D. None of these
11
297 %. Given both o and are noue avec amd sin = 12,
9. Given both O and o are acute angles and sin 0 = 1/2,
cos 0 = 1/3, then the value of 8 + o belongs to
17
.
a 2n
a.
57
3
11
298 8. For the equation 1- 2x – x2 = tan?(x + y) + cot(x + y)
a. exactly one value of x exists
b. exactly two values of x exists
c. y=-1 + nnt + 7/4, ne z
d. y= 1 + nn + /4, ne Z
11
299 A wheel makes 240 revolutions in one
minute The measure of the angle it
describes at the centre in 15 seconds is
( mathbf{A} cdot 60 pi )
в. ( 120 pi )
( c cdot 8 pi )
D. ( pi )
11
300 Write the following relation in the Roster
form and hence find its domain and
range ( R=left{left(a, a^{2}right) / a ) is a prime right.
number less than ( 15} )
11
301 If ( alpha+beta=frac{pi}{2} ) and ( sin alpha=frac{1}{3}, ) then ( sin beta )
is equal to
A. ( frac{sqrt{2}}{3} )
B. ( frac{2 sqrt{2}}{3} )
( c cdot frac{2}{3} )
D. 3
11
302 2. Which of the following statements are always correct
(where Q denotes the set of rationals)?
a. cos 20 € Q and sin 20 € Q tano e Q (if defined)
a b. tan de Q sin 20, cos 20 and tan 20 € Q (if defined)
c. if sin 0 e Q and cos O e Q tan 30 € Q (if defined)
d. if sin 0 e Q cos 30 € Q
11
303 Illustration 3.87 Prove that
sin 10° sin 30° sin 50° sin 70º = 1/16.
11
304 The solution of the equation ( 9 cos ^{12} x+ )
( cos ^{2} 2 x+1=6 cos ^{6} x cos 2 x+ )
( 6 cos ^{6} x-2 cos 2 x ) is/are:
( mathbf{A} cdot mathbf{x}=mathbf{n} pi+frac{pi}{2} ; mathbf{n} in mathbf{I} )
в. ( _{x}=n pi+cos ^{-1}(sqrt[4]{frac{2}{3}}), n in I )
c. ( _{x=n pi-cos ^{-1}}(sqrt[4]{frac{2}{3}}), n in I )
D. none of the above
11
305 28. The equation 2sin? 0+ (21 – 3)sin? -(32+2)sin 0-22
= 0 has exactly three roots in (0, 21), then a can be equal
to
a. 0
b. 2
c. 1
d. -1
11
306 74. tan• ** – 33 tan* * +27 tan? ” is equal to
b. 13
a. 0
c. 3
.00
d. 9
11
307 Prove that ( frac{tan theta}{1-cot theta}+frac{cot theta}{1-tan theta}=1+ )
( sec theta operatorname{cosec} theta )
11
308 15. If logiosin x + log1ocos x = -1 and log10(sin x + cos x) =
(log10 n)-1
-, then the value of ‘n/3′ is
2
11
309 Prove that:
( cos 2 alpha cos 2 beta+sin ^{2}(alpha-beta)-sin ^{2}(alpha+ )
( boldsymbol{beta})=cos 2(boldsymbol{alpha}+boldsymbol{beta}) )
11
310 56. sin 105° + cos 105° will be
equal to
(1) sin 45º (2) tan 45º
(3) cosec 45° (4) sec 45°
11
311 9. ABC is a triangle such that sin(2A + B) = sin(C – A) =
– sin(B + 2C) = 1/2. If A, B and C are in A.P. determine
the values of A, B, and C.
(IIT-JEE 1990)
11
312 Find the value of ( sin left(15^{0}right) )
A ( frac{sqrt{3}-1}{2 sqrt{2}} )
B. ( frac{sqrt{3}+1}{2 sqrt{2}} )
c. ( frac{sqrt{3}-1}{sqrt{2}} )
D. None of these
11
313 If ( A ) is in the 3 rd quadrant and ( B ) is in the
fourth quadrant and ( cos A= ) ( -frac{15}{17}, cos B=frac{4}{5} ) find the value of
( cos (boldsymbol{A}+boldsymbol{B}) )
A ( cdot-frac{84}{85} )
в. ( frac{84}{85} )
( c cdot-frac{34}{85} )
D. None of these
11
314 If ( boldsymbol{x}=boldsymbol{y} cos frac{boldsymbol{2} boldsymbol{pi}}{mathbf{3}}=boldsymbol{z} cos frac{boldsymbol{4} boldsymbol{pi}}{boldsymbol{3}}, ) then ( boldsymbol{x} boldsymbol{y}+ )
( y z+z x ) is equal to
( mathbf{A} cdot-4 x^{2} )
B.
( c cdot 1 )
D.
11
315 ( frac{1+sin 2 theta}{1-sin 2 theta}=left(frac{1+tan theta}{1-tan theta}right)^{b} )
Find ( b )
11
316 34. The total number of solution of sin^x + cos4x = sin x cos x
in [0, 2n) is equal to
a. 2
b. 4
c. 6
d. none of these
11
317 4. Let 2 sinºx + 3 sin x – 2 > 0 and x2 – x – 2 < 0 (x is
measured in radians). Then x lies in the interval
51108 028
b.
c. (-1,2)
IT-JEE 1994)
11
318 The value of ( frac{1}{operatorname{cosec}^{2} theta}+frac{1}{sec ^{2} theta} ) is
( mathbf{A} cdot mathbf{1} )
B.
( c cdot sin ^{2} theta )
( mathbf{D} cdot cos ^{2} theta )
11
319 89. Find the number of pairs of integer (x, y) that satisfy the
following two equations:
cos(xy) = x
tan (xy) = y
b.
2a
od d. 6
0
a. 1
c. 4
11
320 63. The simplest value of cot 9° cot
27° cot 63° cot 81° is
(1) 0
(2) 1
(3) -1 (4) 3
11
321 1. The value of the expression
tan²20° – sin²20°
tan²20°sin20°
is
11
322 Evaluate:
( frac{5 sin ^{2} 30^{0}+cos ^{2} 45^{0}+4 tan ^{2} 60^{0}}{2 sin 30^{0} cos 60^{0}+tan 45^{0}}=frac{55}{m} )
Find ( m )
11
323 64. If cos 40º = a, then value of
cos 100° will be
(1) 1-2a2 (2) 2a2 – 1
(3) 2a +1 (4) 2a
11
324 Let ( A B C ) be a acute angled triangle such that ( A=frac{pi}{3} ) and ( cos B cos C=P )
The possible range of values of ( boldsymbol{P} ) will be
A ( cdotleft(-frac{1}{4}, frac{1}{4}right] )
в. (0,1]
( c cdotleft[frac{1}{3}, inftyright) )
( mathbf{D} cdot[1, infty) )
11
325 1. Which of the following number(s) is/are rational?
are a. sin 15°
b. cos 15°
c. sin 15º cos 15° d. sin 15° cos 75°
11
326 Simplify ( 2 x^{2}+y^{2}+2 x y=5 ) where,
( x=(2 cos theta-sin theta) ) and ( y=(cos theta) )
( 3 sin theta) )
11
327 34. If x, and x2 are two distinct roots of the equation a cos x
+ b sinx = c, then tan ***2 is equal to
a.
b. 6
11
328 The number of values of ( x ) in the interval
( [0,5 pi] ) satisfying the equation ( 3 sin ^{2} x-7 sin x+2=0 ) is
A .
B. 5
( c cdot 6 )
D. 10
11
329 Obtain the value of ( frac{cos 45^{circ}}{sec 30^{circ}+operatorname{cosec} 30^{circ}} ) 11
330 3. The number of distinct solutions of the equation
cos? 2x + cos+ x + sin* x + cos x + sinº x = 2 in the
interval [0, 21) is
(JEE Advanced 2015)
11
331 15. If (cosec?0 – 4)x2 + (cot 8 + V3 ) x + cos2 34 = 0 holds
true for all real x, then the most general values of e can
be given by (ne Z)
a. 2n1 + 111
b. 2nT +
6
c. 2nt +
d. nt
11
332 In which quadrant is ( theta ), if ( sin theta ) is
positive and ( cos theta ) is negative?
( A )
B. I
( c )
D. IV
11
333 The value of the expression ( frac{2left(sin 1^{circ}+sin 2^{circ}+sin 3^{circ}+ldots+sin 89^{circ}right)}{2left(cos 1^{circ}+cos 2^{circ}+ldots+cos 44^{circ}right)+1} )
is
A ( cdot sqrt{2} )
в. ( frac{1}{sqrt{2}} )
( c cdot frac{1}{2} )
( D )
11
334 Illustration 4.24
Solve sin 20+ cos 0 = 0.
11
335 11. If cot + tan 0 = x and sec -cos O = y, then
a. x sin 8. cos 0 = 1 b. sin?o= y cos e
c. (x+y)1/3 + (xy2)1/3 = 1 d. (x+y)2/3 – (xy2)2/3 = 1
11
336 40. Number of solutions of the equation cos+ 2x + 2sin? 2x =
17(cos x + sin x)8, 0<x<2n is
b. 8
c. 10
d. 16
a. 4
11
337 Prove that:
( frac{cos theta}{1-sin theta}+frac{cos theta}{1+sin theta}=2 sec theta )
11
338 71. If sin 0 – cos 0 = 1 and
0< < 90°, then the value of sin
0 + cos 0 is
(3 ja
47
11
339 When ( n ) is an odd natural number other
than ( 1, ) then the value of ( x ) is
This question has multiple correct options
A. ( -pi / 2 )
B.
( c . pi )
D. ( 3 pi )
11
340 If ( A B=-bar{i}-2 bar{j}-6 bar{k}, B bar{C} C=2 bar{i}-bar{j}+ )
( bar{k}, bar{A} C=bar{i}-3 bar{j}-5 bar{k} . ) Then ( angle B=? )
A ( cdot )
[
begin{array}{l}cos ^{-1}(sqrt{frac{40}{41}}) \ text { B } cdot \ qquad begin{array}{l}text { cos }^{1}(sqrt{frac{6}{41}}) \ text { c. } cos ^{-1}left(frac{6}{41}right)end{array} \ text { D. } cos ^{-1}left(frac{62}{63}right)end{array}
]
11
341 Sol. We have n sin x = mcos xl
Draw the graphs of y=nsin x and y=mcos xl.
Ranges of n sin x and micos x are fo, n] and [0, m],
respectively.
Also, period of each of nisin x and micos x is T.
Graphs of functions are as shown in the following figure.
y = nisin x1
Xv = m/cos x B x
to
save

X
Td2
372
27
5:22
Fig. 4.4
From the figure graphs intersect at four points.
Hence, there are four roots of the equation.
For point A, n sin x = m cos x
tan x =
r=tan-1 m
For point B, x = – tan-
For point C, x = 11 + tan-1
For point D, x = 21 – tan!
11
342 If ( A+B=225^{circ}, ) then
( frac{cot boldsymbol{A}}{mathbf{1}+cot boldsymbol{A}} cdot frac{cot boldsymbol{B}}{mathbf{1}+cot boldsymbol{B}}= )
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. ( frac{1}{2} )
11
343 Illustration 3.50 Show that V2 + 12 + 12 + 2 cos 80 =
2 cos 0,0<</16.
11
344 Evaluate ( sin 29^{circ}-cos 61^{circ} ) 11
345 <a<, then
a. 1 + cota
c. 1- cota
2 cota + – is equal to
sina
b. -1 – cota
d. -1 + cota
11
346 Find the number of solutions of the
equations ( (sin x-1)^{3}+(cos x-1)^{3}+(sin x)^{3}= )
( (2 sin x+cos x-2)^{3} ) in ( [0,2 pi] )
11
347 5. The value of tan a tan ſ tan y tand is
a. – 1/3
b. -2
c. 0
d. none of these
11
348 ( cos ^{2} frac{3 pi}{5}+cos ^{2} frac{4 pi}{5} ) is equal to
A ( cdot 4 / 5 )
B. ( 5 / 2 )
c. ( 5 / 4 )
D. ( 3 / 4 )
11
349 22)
41. The number of values of 0 in the interval
IT I
satisfying the equation (v3)** ° = tan^ +2 tan’e is
b. 4
200
c. 0
d. 1
a. 2
11
350 The value of ( tan 75^{circ} ) is
A ( cdot 1+frac{1}{sqrt{3}} )
B. ( 2-sqrt{3} )
( c cdot 2+sqrt{3} )
D. ( 1+sqrt{3} )
11
351 The solution of ( (2 cos x-1)(3+ )
( 2 cos x)=0 ) in the interval ( 0 leq theta leq 2 pi )
is
( mathbf{A} cdot(pi / 3) )
В. ( (pi / 3,5 pi / 3) )
c. ( [pi / 3,5 pi / 3] )
D. none of these
11
352 The value of ( cot 16^{circ} cot 44^{circ}+ )
( cot 44^{circ} cot 76^{circ}-cot 76^{circ} cot 16^{circ} ) is
A . 3
B. ( frac{1}{3} )
( c cdot-frac{1}{3} )
D. – 3
11
353 52. The value of cos y cos
COS X +
52. The value of cosy cos (5. – 2) – cos (7. – y) cosx +
siny cos ( 6 – x) + cosx sin (7. – ») is zero if
a. X=0
b. y = 0
c. x = y
d. nt + y –
(ne Z)
11
354 Assertion
In a triangle ( A B C ) if ( tan A: tan B: )
( tan C=1: 2: 3 ) then ( A=45^{circ} )
Reason
If ( p: q: r=1: 2: 3 ) then ( p=1 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
355 Illustration 4.34
Solve tan 50= cot20.
11
356 66. The value of sec2 12°–
tan2 78°
(1) 0
(2) 1
(4) 3
(3) 2
11
357 +
Illustration 3.73 Prove that
cos 3x cos 5x
cos 7x
cos 9x
sin 2x sin 4x sin 4x sin 6x sin 6x sin 8x’ sin 8x sin 10x
(cosec x) [cosec 2x – cosec 10x]
11
358 Prove that:
( 2 sin ^{2} frac{3 pi}{4}+2 cos ^{2} frac{pi}{4}+2 sec ^{2} frac{pi}{3}=10 )
11
359 15. Solve the equation for x, sin10x + cosi’x = 29 cos4 2x.
16
11
360 35. General solution of tan 0 + tan 40 + tan 70 = tan tan 40
tan 70 is
a. O=nt/12, where ne Z
b. 0=nt/9, where ne Z
c. 0= nt + Tt/12, where ne Z
d. none of these
11
361 8. The value of cos(a + B) is
12
b. ?
25
13
d. none of these
11
362 If OCR
and cos X
sin
then tan x is
() -(4+57)
d (1+17)
11
363 Illustration 3.86 Prove that
cos 20° cos 40° cos 60° cos 80º = 1/16.
11
364 86. The total number of ordered pairs (x, y) satisfying [x] + byl
= 2, sin (ntx{/3) = 1, is equal to
a. 2
b. 3
d. 6
c. 4
11
365 ( operatorname{Prove} sin left(mathbf{6 0}^{0}+boldsymbol{theta}right)-sin left(mathbf{6 0}^{mathbf{0}}-boldsymbol{theta}right)= )
( sin boldsymbol{theta} )
11
366 Illustration 2.6 Given that sin 30º = 1/2 and cos 30º =
V3/2. Determine the values of sin 60°, sin 120°, sin 240°,
sin 300°, and sin (-30%).
11
367 ( frac{cos 15^{circ}-sin 15^{circ}}{cos 15^{circ}+sin 15^{circ}}=frac{1}{sqrt{3}} ) 11
368 Prove that
[
begin{array}{l}
(cos alpha+cos beta)^{2}+(sin alpha+sin beta)^{2}= \
4 cos ^{2}left(frac{alpha-beta}{2}right)
end{array}
]
11
369 Find the degree measure of the angle
subtended at the centre of a circle
of radius ( 100 mathrm{cm} ) by an arc of length 22
( mathrm{cm} .left(boldsymbol{U s e} boldsymbol{pi}=frac{mathbf{2 2}}{mathbf{7}}right) )
11
370 If ( 1+sin x+sin ^{2} x+sin ^{3} x+ldots infty ) is
equal to ( 4+2 sqrt{3}, 0 leq x<pi ) then ( x ) is
equal to
A.
в.
c. ( frac{pi}{3} ) or ( frac{pi}{6} )
D. ( frac{pi}{3} ) or ( frac{2 pi}{3} )
11
371 11. Number of solutions of the equation sinºx – cos²x sinx
+ 2 sin x + sinx = 0 in 0 SXS 31 is
11
372 Illustration 2.55 If 0 < a < p < y< 1/2, then prove that
sin a + sin ß + sin y
tan a<
cosa + cos ß + cos y
– <tan y.
11
373 The set of angles between ( 0 & 2 pi )
satisfying the equation ( 4 cos ^{2} theta ) ( 2 sqrt{2} cos theta-1=0 ) is –
A ( cdotleft{frac{pi}{12}, frac{5 pi}{12}, frac{19 pi}{12}, frac{23 pi}{12}right} )
В ( cdotleft{frac{pi}{12}, frac{7 pi}{12}, frac{17 pi}{12}, frac{23 pi}{12}right} )
c. ( left{frac{5 pi}{12}, frac{13 pi}{12}, frac{19 pi}{12}right} )
D. ( left{frac{pi}{12}, frac{7 pi}{12}, frac{19 pi}{12}, frac{23 pi}{12}right} )
11
374 26. If x, y, z are in A.P, then
sin x -sin z
is equal to
a. tany
c. siny
COS Z – cos x
b. coty
d. cos y
11
375 Illustration 2.42 If sin?, + sin²0, + sin’ex = 0, then which
of the following is not the possible value of cose + cos O2
+ cos Oz?
a. 3
b. -3
c. -1
d. -2
11
376 65. If cosx = 2-cos y
2 cos y-1
where x, y E (0, ), then tan – cot
is equal to
b. 3
11
377 Illustration 4.40
Solve 2 sin’x + sin2x = 2.
11
378 If ( tan alpha=frac{1}{7}, tan beta=frac{1}{sqrt{10}}, ) prove that
( alpha+2 beta=frac{pi}{a}, ) where ( 0<alpha<frac{pi}{2} ) and
( mathbf{0}<boldsymbol{beta}<frac{boldsymbol{pi}}{mathbf{2}} )
Find ( a )
11
379 If ( sin Theta+cos Theta=sqrt{2}, ) and ( Theta ) is actual
then ( tan Theta ) is equal to
A ( cdot frac{1}{sqrt{3}} )
B.
( c cdot sqrt{3} )
D. ( infty )
11
380 the fundamental period, if any, of the
functions:
( sin (x / 3) ) is ( k pi, ) then ( k ) is
11
381 7.
The general solution of
sin x-3 sin 2x + sin 3x = cos x – 3 cos 2x + cos 3x is
(1989-2 Marks)
(a) n+
(d)
2ntt + cos!
11
382 Illustration 4.49 If x, y e [0, 271), then find the total
number of ordered pairs (x, y) satisfying the equation
sinx cos y = 1.
11
383 69. If roots of the equation 2×2 – 4x + 2 sin 0 – 1 = 0 are of
opposite sign, then e belongs to
BULEVY
(1 51
a. (76
137
177
c. (1991)
d. (0,7)
d. (0, 1)
66
11
384 ( frac{tan 5 Theta+tan 3 Theta}{tan 5 Theta-tan 3 Theta}=4 cos 2 Theta cos 4 Theta ) 11
385 ( cos 1^{circ} cos 2^{circ} cos 3^{circ} dots dots dots dots dots cos 90^{circ} )
( mathbf{A} cdot mathbf{0} )
B . -1
( mathbf{c} cdot 1 )
D. None of these
11
386 prove ( rightarrow frac{1+cos Theta+sin Theta}{1+cos Theta-sin Theta}=frac{operatorname{coz} Theta}{1-sin Theta} ) 11
387 47. Ifcos(a-B)=3 sin(a+B), then –
1-3 sin 2a
1-3 sin 2B
Tail
d.
11
388 Prove the following statements:
( cos ^{4} boldsymbol{A}-sin ^{4} boldsymbol{A}+mathbf{1}=boldsymbol{2} cos ^{2} boldsymbol{A} )
11
389 The value of ( sin ^{2} 20^{circ}+sin ^{2} 70^{circ} ) is :
( mathbf{A} cdot mathbf{1} )
B. 0
( c cdot 2 )
D. 3
11
390 12. The number of solutions of 12 cos’x – 7 cos²x + 4 cosx =
9 is
a. 0
b. 2
c. infinite
d. none of these
11
391 5. Let f(x) = x² – 2/(sin 13 — sin V2) x – (cos 13 – cos 12)
then
a. f(x) is positive VxER
b. f(x) assumes both positive and negative values
c. f(x) = 0 has no real roots
d. y=f(x) touches the line y = 0.
11
392 Illustration 3.34 Prove that
cos A + cos B ( sin A +sin B
sin A -sin B (cos A – cos B
B
A
in
= 2 cot”
OF O,
2
accordingly as n is even or odd.
11
393 95. If tan x = n tan y. ne Rt then the maximum value of
sec-(x – y) is equal to
95. yn
myne P, then the maximum value of
b
8
a. (n+1)2
(n+1)
2
2n
b. (n+1)
c. non
(n+1)
tie 2
d. (n+12
4n
11
394 Simplify ( frac{5 cos ^{2} 60^{0}+4 sec ^{2} 30^{0}-tan ^{2} 45^{0}}{sin ^{2} 30^{0}+cos ^{2} 30^{0}} )
( A cdot-frac{55}{2} )
в. ( frac{67}{12} )
c. ( frac{5}{12} )
D. ( frac{55}{4} )
11
395 ( sin ^{-1} x+sin ^{-1}left(frac{1}{x}right)+cos ^{-1} x+ )
( cos ^{-1}left(frac{1}{x}right)= )
( mathbf{A} cdot pi )
( B cdot frac{pi}{2} )
( c cdot frac{3 pi}{2} )
D. ( 2 pi )
11
396 Illustration 2.17 By geometrical interpretation, prove that
tan a + tan B
tan(a+B) = 1
1- tan a tan ß
11
397 The inequality ( 2^{sin theta}+2^{cos theta} geq 2^{1-(1 / sqrt{2})} )
holds for all real values of ( theta )
A . True
B. False
11
398 84. If A + B + C = 37/2, then cos 2A +cos 2B + cos2C is equal
to
a. 1 – 4 cos A cos B cos C
b. 4 sin A sin B sin C
c. 1 + 2 cos A cos B cos C
d. 1 -4 sin A sin B sin C
11
399 Illustration 4.32
Solve tan 30=-1.
11
400 Evaluate each of the following:
( frac{sin 60^{circ}}{cos ^{2} 45^{circ}}-cot 30^{circ}+15 cos 90^{circ} )
11
401 61. The value of cot 70° + 4 cos 70° is
b. √3
c. 2/3
d1 0800
2
11
402 The number of roots of the equation ( x+2 tan x=frac{pi}{2} ) in the interval ( [0,2 pi] ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 3 )
D. Infinite
11
403 Solve for ( x ) and ( y: ) ( boldsymbol{x}+boldsymbol{y}=frac{boldsymbol{7} boldsymbol{pi}}{boldsymbol{4}}, frac{sin boldsymbol{x}}{cos boldsymbol{y}}=-sqrt{boldsymbol{2}} ) 11
404 Find the value of:
( 2 tan 45^{circ}+cos 45^{circ}-sin 45^{circ} )
11
405 ( fleft(x-frac{x^{2}}{4}+frac{x^{3}}{4}-dots dots inftyright)+ )
( cos ^{-1}left(x^{2}-frac{x^{4}}{4}+frac{x^{6}}{4}-ldots ldots . . inftyright)=frac{pi}{2} )
and ( 0<x<sqrt{2} ) then ( x= )
A ( cdot frac{1}{2} )
B.
( c cdot-frac{1}{2} )
D. –
11
406 9.
For a positive integer n, let
(1999 – 3 Marks)
CO
1+ sec ) (1+sec 20) (1 + sec 40)….(1 + sec 2″).
Then
o sa (5)=1
@ ss (153)=1
11
407 Illustration 3.11 Let a, Band y satisfy 0<a<B Y 2 .1
cos (x + a) + cos (x + B) + cos (x + y) = ( for all x ER, then
find the possible values of (Y-a).
11
408 ( frac{sin (B-C)}{cos B cos C}+frac{sin (C-A)}{cos C cos A}+ )
( frac{sin (A-B)}{cos A cos B}= )
11
409 Convert ( 25^{circ} ) into radian. 11
410 Illustration 4.60 Prove that the least positive value of x,
satisfying tan x = x + 1, lies in the interval (71/4, 1/2).
11
411 ( operatorname{Prove} cos ^{2} alpha-sin ^{2} 2 alpha= )
( cos ^{2} alpha cos 2 alpha-2 sin ^{2} alpha cos ^{2} alpha )
11
412 The measure of an angle in degrees, grades and radians be ( mathrm{D}, mathrm{G} ) and ( mathrm{C} ) respectively, then relation between them ( frac{D}{90}=frac{G}{100}=frac{2 C}{pi} ) but ( 1^{circ}= )
( left(frac{180}{pi}right)^{circ} simeq 57^{circ}, 17^{prime}, 44.8^{prime prime} ) and sum of
interior angles of a ( n ) -sided regular polygon is ( (2 n-4) frac{pi}{2} . ) On the basis of above information, answer the following questions :Which of the following are correct
This question has multiple correct options
( A cdot sin 1^{circ}cos 1^{circ} )
( mathbf{c} cdot cos 1^{circ}<cos 1^{c} )
D. ( 1^{c}=57^{circ} Rightarrow sin 1^{c}cos 57 )
11
413 The value of ( 9 tan ^{2} theta-9 sec ^{2} theta ) is
( mathbf{A} cdot mathbf{1} )
B. 0
( c .9 )
D. – –
11
414 Which quadrant ( 180<theta<270 )
degrees lies?
A . quadrant I
B. quadrant III
c. quadrant!
D. quadrant IV
11
415 If ( cos left(frac{3 pi}{4}+xright)-cos left(frac{3 pi}{4}-xright)= )
( -sqrt{m} sin x . ) Find ( m )
11
416 12. The expression cos(a + b + cos-(a – b) – cos 20.
cos 2ß, is
a. independent of a
b. independent of ß
c. independent of a and ß
d. dependent on a and ß.
11
417 If ( x=cos 10^{circ} cos 20^{circ} cos 40^{circ}, ) then the
value of ( x ) is
A ( cdot frac{1}{4} tan 10^{circ} )
B. ( frac{1}{8} cot 10^{circ} )
c. ( frac{1}{8} cos 10^{circ} )
D. ( frac{1}{8} sec 10^{circ} )
11
418 air of equations
4. The number of solutions of the pair of eau
2sin? 0 – cos2 0 = 0 and 2 cos? 0 – 3sin 0 = 0) in
interval [0, 21) is
a. O
b. 1
c. 2
d. 4
11
419 77. The equation sinºx – 2cos x + a2 = 0 can be solved if
a. – √3 sas ſ3 b. -√2 sas √2
c. -1 Sas1
d. None of these
11
420 Express the following angle into radians
( mathbf{5 0}^{circ} mathbf{3 7}^{prime} mathbf{3 0}^{prime prime} )
11
421 If ( tan theta=frac{b}{a}, ) then the value of
( a cos 2 theta+b sin 2 theta ) is
( mathbf{A} cdot b )
в.
c. ( frac{a}{b} )
D. ( frac{a}{a+b} )
11
422 Illustration 3.90
Prove that
147
2n 40
8T
COSCOS – COS – COS
15
15
15
15
16
11
423 The value of ( frac{7 pi^{circ}}{9} ) in sexagesimal measure is
A ( .120^{circ} )
B. ( 130^{circ} )
c. ( 140^{circ} )
D. ( 150^{circ} )
11
424 If ( A=60^{circ} ) verify the following.
( sin 2 A=2 sin A . quad cos A )
11
425 87. The value of
3 IN
cos
is equal to
r=0
a. 1/4
c. -1/4
b. 1/8
d. -1/8
11
426 7.
The number of values of x in the interval [0,3Tt] satisfying
the equation 2 sin2 x+5 sin x-3=0 is
[2006]
(2) 4 (6)
6 C 1 (d) 2
11
427 5. If sin 0 – cos 0 = 1, then the value of sine – cos O is 11
428 The value of ( tan 105^{circ} ) is
A. ( -(2+sqrt{3}) )
B. ( -2+sqrt{3} )
c. ( sqrt{3}+sqrt{2} )
D. ( -(sqrt{3}+sqrt{2}) )
11
429 Find the value of ( operatorname{cosec}^{2} 60^{circ}-tan ^{2} 30^{circ} ) 11
430 tan? T
tan 2
.
*+tan2 31
11. The value of —
cot??
& cot2 20
7
+ cot 30
a. 7
c. 21/5
b. 35/3
d. none of these
11
431 16. The value of
sin 1° + sin 3° + sin 5° + sin 7°
cos 1°•cos 2°•sin 4°
11
432 Find the degree measure of ( frac{pi}{8} ) 11
433 1. Solve 3 tan 2x – 4 tan 3x = tan²3x tan 2x. 11
434 The value of ( sin 50^{circ}-sin 70^{circ}+sin 10^{circ} )
is
A . 0
B.
( c cdot frac{1}{2} )
D. ( frac{1}{sqrt{2}} )
11
435 4 ST
Illustration 3.54
Prove that cos*
+ cos
+ cost da
+ cos4
NIw
8
11
436 tan
2
58. What will be the value of
tan 6-a) from the follow-
ing?
(1)
(1) 1+ sin A
COSA 1-sin A
(2) cos A
(3) Both of above
(4) None of the above
11
437 Illustration 4.48 For what value of k the equation sin x +
cos(k +x) + cos(k – x) = 2 has real solutions?
11
438 1. Find the value of sin (105°).
10
11
439 Find the general solution of each of the following equations. ( sin left(x+frac{pi}{5}right)=0 ) 11
440 If ( cos ^{2} x+cos ^{2} 2 x+cos ^{2} 3 x=1, ) then
A ( x=(2 n+1) frac{pi}{4}, n in I )
в. ( _{x}=(2 n+1) frac{pi}{2}, n in I )
c. ( _{x=n pi pm} frac{pi}{6}, n in I )
D. None of these
11
441 Convert 4 radians into degree measure
and also convert ( -47^{circ} 30^{prime} ) into radian
measure.
11
442 18. If cos 30= os 3 a, then the value of sin
can be given by
(T
a.
sin a
b. sin
– ta
3
(
3
11
443 Solve: ( frac{cos ^{2} 25^{circ}+cos ^{2} 65^{circ}}{sin ^{2} 59^{circ}+sin ^{2} 31^{circ}}=? )
( mathbf{A} cdot mathbf{0} )
B.
c. 2
( D )
11
444 Find radian measure corresponding to
the degree measure ( -37^{circ} 30^{prime} )
11
445 1-tan-
Illustration 3.43
Prove that
-= sin 2 A.
1+tan?
– A)
11
446 If ( operatorname{cosec} theta-cot theta=p, ) then ( operatorname{cosec} theta+ )
( cot theta= )
( A cdot 1 / p )
B. -1/p
( c cdot-p )
D ( cdot p^{2} )
11
447 1. If cos(A – B) = 3/5 and tan A tan B = 2, then
a. cos A cos B =1/5 b. sin A sin B = -2/5
c. cos A cos B=-1/5 d. sin A sin B =-1/5
11
448 If ( f(x)=left[cos x cos (x+2)-cos ^{2}(x+right. )
1) ( ] ) where ( [.] ) denotes the greatest integer function ( leq x ). Then solution of
the equation ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} ) is :
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. none of these
11
449 53. The value of sinº 10° + sinº 50°– sinº 70° is equal to
b. A
4
11
450 14. Number of triangles ABC if tan A = x, tan B = x + 1, and
tan C= 1- x is
11
451 UCH
74.
If sec x + cos x = 2, then the
value of sec16 x + cos 16 x will be
(1) 13 (2) 2
(3) 1
(4) O
11
452 The measure of an angle in degrees,
grades and radians be ( mathrm{D}, mathrm{G} ) and ( mathrm{C} )
respectively, then relation between them ( frac{boldsymbol{D}}{mathbf{9 0}}=frac{boldsymbol{G}}{mathbf{1 0 0}}=frac{boldsymbol{2} boldsymbol{C}}{boldsymbol{pi}} ) but ( mathbf{1}^{circ}= )
( left(frac{180}{pi}right)^{0} simeq 57^{circ}, 17^{prime}, 44.8^{prime prime} ) and sum of
interior angles of a ( n ) -sided regular polygon is ( (2 n-4) frac{pi}{2} . ) On the basis of above information, answer the following questions:One angle of a triangle is ( frac{4 x}{3} )
grades and another is ( 3 x ) degrees, while the third is ( frac{2 pi x}{75} ) radians. Then the angles in degrees are –
A ( cdot 20^{circ}, 60^{circ}, 100^{circ} )
B . 24 ‘ , 60 ( ^{circ}, 96^{circ} )
c. ( 36^{circ}, 60^{circ}, 84^{circ} )
D. ( 20^{circ}, 40^{circ}, 120^{circ} )
11
453 If ( sqrt{frac{1-sin A}{1+sin A}}+frac{sin A}{cos A}=frac{1}{cos A}, ) for all
permissible values of ( A, ) then ( A ) belongs
to
This question has multiple correct options
A. First Quadrant
B. Second Quadrant
c. Third Quadrant
D. Fourth Quadrant
11
454 17. The value of x in (0, TT/2) satisfying –
3-
13+1
sin x cos x
= 4-2 is
11
455 Illustration 4.46 Find the number of integral values of n so
that sinx (sinx + cos x) = n has at least one solution.
11
456 16. If sin (sin x + cos x) = cos (cos x – sin x), and largest
possible value of sin x is
, then the value of k is
11
457 Sector area of a circle in radians is
A ( cdot 2 times frac{theta}{2} r^{2} )
в. ( frac{theta}{2} times r^{2} )
c. ( _{pi times frac{theta}{2} r^{2}} )
D. ( _{2 times frac{theta}{2} pi r} )
11
458 If the angles a, b, y of a triangle satisfy the relation,
2222”
16. The measure of the smallest angle of the triangle is
a. 30°
b. 40°
c. 450
d. 50°
11
459 11. If sin(x + 20°) = 2 sinx cos 40º, where x € (0, 1/2), then
which of the following hold(s) good?
a. cos 2x = 1/2 b. cosec 4x = 2
c. sec * = V6 – v2 d. tan* = (2-13)
11
460 Greatest possible difference between two of the roots if ( boldsymbol{theta} boldsymbol{epsilon}[mathbf{0}, boldsymbol{2} boldsymbol{pi}] ) is
A . 2
B.
( c cdot sqrt{2} )
D. ( 2 sqrt{2} )
11
461 25. Let 0 € 0, and t; = (tane)tano, tz = (tane)coto,
tz =(coto) tane and t4 =(coto) cote, then (2006 – 3M, -1)
(a) 4 >t2>tz > 14
(b) to> tz>t>tz
c) tz >t>t2>14 (d) t2>tz >t>t4
3
1
11
462 19. x = Va’ cosa + b2 sin’ a + Va? sina + b2 cos? a
then x2 = a + b2 +2 Vp (a? +b?) – p, where p is equal
to
a. dcos?a + b² sinʼa
af sin’a + b2 cosa
[a? + b2 + (a? – b?) cos 2a]
[a? + b2 – (a? – 62) cos 2a]
11
463 55. If tanda = 1 + 2tanB, then
(90°-2a)] will be equal
(1) cos B
(2) 1 + 2cos
(3) 1 + cos28
(4) 2 cosa
11
464 54. If tan 0-3sino = 0 then what
will be the value of sin’e –
coso –
(1) (1672 +1)
(2) 27 (16/2 –1)
(3) (1672 – 1)
(4) 2 (1672 +1)
11
465 Find the value of ( sin 60^{circ} ) geometrically. 11
466 The value of ( frac{1}{cos 290^{circ}}+frac{1}{sqrt{3} sin 250^{circ}} ) is?
( A cdot frac{2 sqrt{3}}{3} )
B. ( frac{4 sqrt{3}}{3} )
( c cdot sqrt{3} )
D. None
11
467 Find the degree measure of ( frac{1^{c}}{4} ) 11
468 11. For a triangle ABC it is given that cos A+ cos B+cos C =
Prove that the triangle is equilateral.
11
469 Prove the following identities. ( frac{1+sin alpha}{1+cos alpha} cdot frac{1+sec alpha}{1+operatorname{cosec} alpha}=tan alpha ) 11
470 The value of ( cos (A+B) ) if ( sin A=frac{3}{5} )
and ( cos B=frac{8}{17} )
11
471 8. The number of integral values of k for which the equation
7 cos x + 5 sin x = 2k + 1 has a solution is
a. 4
b. 8
c. 10
d. 12 (IIT-JEE 2002)
11
472 The complete set of values of ( x )
satisfying equation ( cot x-cos x= )
( 1-cot x cos x ) is
A ( cdotleft{x: x=(4 n pi+1) frac{pi}{4}, n in Iright} )
в. ( left{x: x=2 n pi+frac{pi}{4}, n in Iright} )
C ( cdot{x: x=2 n pi pm pi, n in I} )
D ( cdot{x: x=2 n pi+pi, n in I} cupleft{x: x=n pi+frac{pi}{4}, n in Iright} )
11
473 ( boldsymbol{y}=sec ^{-1}left(frac{1}{2 x^{2}-1}right) )
Prove that: ( boldsymbol{y}=2 cos ^{-1} boldsymbol{x} )
11
474 5. Solve the equation tan^x + tan^y + 2 cot?x cotły =
3+ sin? (x + y) for the values of x and y.
11
475 Illustration 2.33 If sinf a + cos4 B + 2 = 4 sin a cos B,
osa, ßs then find the value of (sin a + cos 3).
11
476 If ( tan A, tan B ) are the roots of ( x^{2}- )
( mathbf{2} boldsymbol{x}+mathbf{2}=mathbf{0} ) then ( cot ^{2}(boldsymbol{A}+boldsymbol{B})= )
A ( cdot frac{4}{5} )
в. ( frac{1}{2} )
( c cdot frac{3}{4} )
D.
11
477 s of the equation
18. If sum of all the solutions of the
8c08* (cos+x.cos(x) 7.)-1 in (0, 1) iska
COS
[JEE M 2018)
then k is equal to :
11
478 If ( tan theta+tan phi=a, cot theta+cot phi= )
( boldsymbol{b}, boldsymbol{theta}-boldsymbol{phi}=boldsymbol{alpha} neq mathbf{0}, ) then
This question has multiple correct options
( mathbf{A} cdot a b>4 )
в. ( a b<4 )
( ^{mathbf{c}} tan ^{2} alpha=frac{a b(a b-4)}{(a+b)^{2}} )
D ( sec ^{2} alpha=frac{(a-b)^{2}+a^{2} b^{2}}{(a+b)^{2}} )
11
479 A ladder makes an angle ( 30^{circ} ) with the
floor and its lower end is 12 m away
from the wall. Find the length of the ladder
11
480 ( frac{sin 30^{circ}}{sin 45^{circ}}+frac{tan 45^{circ}}{sec 60^{circ}}-frac{sin 60^{circ}}{cot 45^{circ}}-frac{cos 30^{circ}}{sin 90^{circ}} ) 11
481 44. If
<a<3
then
1- cos a
V1+cos a
1+cos a
– is equal to
11- cos a
2
sin a
00
a.
b
.

sin a
1
c.
sin a
R
sae d. —
o llosin a
lot
11
482 14. Ifp = sin (A – B) sin (C-D), q = sin (B-C) sin (A-D),
r=sin (C – A) sin (B-D) then
a. p +q-r=0 b. p + q + r = 0
c. p-+r=0
d. p3 +93 +p3 = 3pqr
11
483 If ( A+B+C=180^{circ} ) then ( a sin (B-C)+b sin (C- )
( A)+c sin (A-B) ) is :
( A cdot O )
B. 1
( c cdot 2 )
D. 3
11
484 Evaluate ( cos (A+B) cdot cos (A-B)= )
( A cdot sin ^{2} A-cos ^{2} B )
B. ( cos ^{2} A+sin ^{2} B )
( mathbf{c} cdot cos ^{2} A-sin ^{2} B )
( mathbf{D} cdot cos ^{2} A+cos ^{2} B )
11
485 27. The value of f(x) = x4 + 4×3 + 2×2 – 4x + 7, when x =

110
cot –
E is
11
486 If ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi} ) and ( mathbf{A}, mathbf{B}, mathbf{C} ) are all
greater then 0 and angle ( C ) is obtuse then
( mathbf{A} cdot tan A tan B geq 1 )
B. ( tan A tan B<1 )
( mathbf{c} cdot tan A tan B=1 )
D. None of these
11
487 23.
cos(a-B)= 1 and cos(a+B) = 1le where a Ber-.
Pairs of a, ß which satisfy both the equations is are
(20055
(a) 0 (b) 1 (c) 2 0 @ 4
11
488 Illustration 3.101 Find the least value of sec A + sec B +
sec C in an acute angled triangle.
11
489 If ( A+B=45^{circ}, ) prove that ( (cot A- )
1) ( (cot B-1)=2 )
11
490 llustration 3.53
lustration 3:53 Prove that tan mo +2 tan +4 = com o
Prove that tan
Kloo
16
11
491 Prove that ( : frac{1+cos A+sin A}{1+cos A-sin A}= )
( frac{1+sin A}{cos A} )
11
492 40. If 2 sec 20 = tan o+cot o, then one of the values of 0+ 0
is
a. 1/2
c. 7/3
b. a /4
d. none of these
11
493 Let a,ß be such that i<a-B<37.
If sin a + sin B =-
and cos a + cos B = —
then the
65
value of cosa,
[2004]
(6) Viso
11
494 34. The set of values of le R such that sin? 0+ cos 0= 2 cosa e
holds for some 8, is
a. (-∞, 1]
b. (-0, -1]
c. 0
d. [-1,-)
11
495 If ( sin theta=frac{45}{53}, ) find the value of
( operatorname{cosec}^{2} theta-cot ^{2} theta )
11
496 What is the value of ( frac{sin ^{2} 30^{0}}{cos ^{2} 30^{0}}+ )
( frac{cos ^{2} 30^{0}}{sin ^{2} 30^{0}} ? )
11
497 30. The minimum value of a tan²x + b cot?x equals the
maximum value of a sin’e + b cos?where a > b > 0.
Then alb is
a. 2
b. 4
c. 6
d. 8
11
498 Illustration 2.37 Which of the following is the greatest?
a. tan 1
b. tan 4
c. tan 7
d. tan 10
11
499 Illustration 3.44 Prove that (cos A – cos B)2 + (sin A – sin B)?
= 4 sin’[(A – B)/2].
11
500 sin 2x
26. The least positive solution of cot
lies in
11
501 If ( 3 x=sec theta ) and ( frac{3}{x}=t a n theta, ) then find the value of ( 9left(x^{2}-frac{1}{x^{2}}right) ) 11
502 If ( alpha, beta ) are two different values of ( theta ) lying
between 0 and ( 2 pi ) which satisfy the
equation ( 6 cos theta+8 sin theta=9 ) Find
( cos (alpha+beta) )
11
503 Solve for ( x, sin ^{2} 2 x=(sin 2 x) ) 11
504 3. 3 (sin e-cos )4 + 6 (sin 0 + cos 02 +4 (sin e + cos®) is
equal to
a. 11
c. 13
b. 12
d. 14
(IIT-JEE 1995)
11
505 Illustration 4.23 Solve V3 sec 20= 2. 11
506 For each natural number ( k ), let ( C_{k} )
denote the circle with radius ( k )
centimeters and centre at the origin ( O )
on the circle ( C_{k} ) a particle moves ( k ) centimetres in the counter-clockwise
direction. After completing its motion
on ( C_{k}, ) the particle moves to ( C_{k+1} ) in the radial direction (away from centre). The motion of the particle continues in this manner. The particle starts at ( (1,0) . ) If the particle crosses the positive direction of ( x ) -axis for the first time on
the circle ( C_{n}, ) then ( n= )
A. 4
B. 5
c. 6
D.
11
507 cos(A+C)
69. If cos 2 B=
cos(A-C)
2, then tan A, tan B, tan C are in
a. A.P.
b. G.P.
c. H.P.
d. none of these
11
508 Illustration 3.55
If it < x < 21, prove that
V1 + cos x + VI – cos x
V1 + cos x – V1 – cos x
== cot
+
Rt
11
509 If in a ( Delta A B C, cos A . cos B+ )
( sin A cdot sin B cdot sin C=1, ) then triangle
( A B C ) is
A. isosceles
B. right angled
c. equilateral
D. right angle isosceles
11
510 In a ( triangle A B C, ) the angles ( A ) and ( B ) are two
different values of ( theta ) satisfying
( sqrt{mathbf{3}} cos theta+sin theta=k,|k|<2 . ) The
triangle:
A. is an acute angled
B. is aright angled
c. is an obtuse angled
D. has one angle ( =frac{pi}{3} )
11
511 26. If cotA cot?B = 3, then the value of (2 – cos 2A) (2 –
cos 2B) is
11
512 29. If 2 sin? ((1/2) cos2x) = 1 – cos(a sin 2x),x + (2n+1) 7/2,
ne I, then cos 2x is equal to
a. 1/5
b. 3/5
c. 4/5
d. 1
11
513 51. For n e Z, the general solution of (V3 – 1) sin 0 + ( 13
+ 1) cos 0 = 2 is (ne 2)
a. 0 = 2n1 = ” +
b. 0= nt+ (-1)”
c. 6=27=
c. O= 2nt
+
BIBIT
1
d.=na+(1)I
11
514 Illustration 2.4 If 3 sin 0 + 5 cos 0 = 5, then show that
5 sin – 3 cos O=+3.
11
515 If ( cos x=sqrt{1-sin 2 x}, 0 leq x leq pi, ) then
possible value of ( x ) is
This question has multiple correct options
A . ( pi )
B.
( c cdot tan ^{-1} 2 )
D. ( 3 pi )
11
516 What is the simplified value of ( frac{sin 2 A}{1+cos 2 A} ? )
( mathbf{A} cdot tan A )
B. ( cot A )
( c cdot sin A )
D. ( cos A )
11
517 The value of ( x ) between 0 and ( 2 pi ) which
satisfy the equation ( sin x sqrt{8 cos ^{2} x}=1 )
are in A.P. The common difference of the
A.P. is
A ( cdot frac{pi}{8} )
B. ( frac{pi}{4} )
c. ( frac{3 pi}{8} )
D. ( frac{5 pi}{8} )
11
518 A chord of a circle of radius ( 12 mathrm{cm} )
stands an angle of ( 120^{circ} ) at the centre.
Find the area of the corresponding
segment of the circle.
11
519 Prove the following identities:
( 1+cos ^{2} 2 x=2left(cos ^{4} x+sin ^{4} xright) )
11
520 42. If the inequality sin²x + a cos x + a² > 1 + cos x holds for
any x e R, then the largest negative integral value of a is
a. 4
b. -3
c. -2
d. -1
11
521 If ( sin ^{x} alpha+cos ^{x} alpha geq 1,0<alpha<frac{pi}{2}, ) then
A. ( x in[2,+infty) )
в. ( x in(-infty, 2] )
c. ( x in[-1,1] )
D. none of these
11
522 Find the value of ( 4 cos 60^{circ}+2 sin 30^{circ} )
( mathbf{A} cdot mathbf{0} )
B . 2
( c .-3 )
D. 3
11
523 3. The number of values of x in the interval [0,57] satisfying
the equation 3 sin’x – 7 sin x + 2 = 0 is
b. 5
d. 10 (IIT-JEE 1998)
a. 0
c. 6
11
524 Prove that ( frac{1}{sin 10^{circ}}-frac{sqrt{3}}{cos 10^{circ}}=4 ) 11
525 ( sin theta=frac{3}{5}, operatorname{cosec} theta=? ) 11
526 • The value of the expression (2 sin2 91°- 1) (2 sin? 92°- 1)
… (2 sind 180° – 1) is equal to
a. 0
d. 290 – 90
b. 1
c. 290
11
527 Find general solution of the following equations:
( sin theta=frac{1}{2} ? )
11
528 Solve: ( sqrt{sec x-1}=tan x ) 11
529 1. The number of all the possible triplets (ay, ay, ay) such that
a, + a2 cos(2x) + az sin*(x) = 0 for all x is
a. O
b. 1
c. 3
d. infinite (IIT-JEE 1987)
11
530 If ( sin A=x, cos B=y ) and ( A+B+ )
( C=0, ) then ( x^{2}+2 x y sin C+y^{2} ) is
equal to
( mathbf{A} cdot sin ^{2} C )
B. ( cos ^{2} C )
c. ( 1+sin ^{2} C )
D. ( 1+cos ^{2} C )
11
531 4. If x, y e R satisfies (x + 5)2 + (y – 12)2 = (14)2, then the
minimum value of Vx² + y2 is_
2
11
532 Prove that: ( cos ^{2} A+cos ^{2} B+cos ^{2} C= )
( 1-2 cos A cos B cos C )
11
533 6. (a + 2) sin a + (2a – 1) cos a=(2a + 1) if tan a is
a. 3/4
b. 4/3
c. 2a/(a + 1)
d. 2al(a? – 1)
11
534 If ( sec theta+tan theta=p )
Find ( csc =? )
11
535 Illustration 4.9 Find the number of solutions of sinºx – sinx
– 1 = 0 in [-216, 210).
11
536 ( mathrm{IF} sin Theta=frac{8}{17} ) where ( 0^{circ}<Theta<90^{circ} )
then ( tan Theta+sec Theta ) is
( A cdot frac{1}{3} )
B. ( frac{2}{3} )
( c cdot frac{4}{3} )
D.
11
537 The expression ( 2 cos 10^{circ}+sin 100^{circ}+ )
( sin 1000^{circ}+sin 10000^{circ} ) is simplified
then it simplifies to?
11
538 ( tan left(rho frac{pi}{4}right)=cot left(q frac{pi}{4}right) )
A ( . rho+q=0 )
В ( cdot rho+q=2 n+1 )
c. ( rho+q=2 n )
D ( cdot rho+q=2(2 n+1) )
11
539 Convert ( frac{7 pi}{36} ) into degrees. 11
540 If ( tan theta=frac{1}{sqrt{5}} ) and ( theta ) lies in the first
quadrant, the value of ( cos theta ) is
A ( cdot frac{1}{sqrt{6}} )
B. ( frac{sqrt{5}}{sqrt{6}} )
c. ( frac{-1}{sqrt{6}} )
D. ( frac{-sqrt{5}}{sqrt{6}} )
11
541 7. Show that 16cos( 25 cos ( 16 cos ( 15 cos 165 = 1
CC
(1983 – 2 Marks)
11
542 5.
Let 2sin²x + 3sinx-2> 0 and x2-x-2<0 (x is measured in
radians). Then x lies in the interval
(1994)
(c) (-1,2)
11
543 Q Type
elevation of the top ( P ) of the tower ( O P ) at
a point ( A ) on the ground is ( alpha, ) he then
walks a distance ( A B ) towards the foot ( O )
of the tower and finds the angle of
elevation as ( beta, ) he again walks a
distance ( B C ) in the same direction and
observes the angle of elevation now is ( gamma )
He notices that ( boldsymbol{a}+boldsymbol{beta}+boldsymbol{gamma}= )
( 180^{circ} ; alpha, beta, gamma ) are in A.P. and the distance
of ( C ) from the foot of the tower is half the
distance of ( B ) from the foot ( O ) of the
tower. If the height of tower is ( h ), the distance of ( A ) from the foot ( O ) is ( frac{5 h}{3 sqrt{3}} )
Reason

If the angles of elevation of the top of a
tower at three points on the ground are in A.P. then the distance of the points from the foot of the tower are also in A.P.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect

11
544 ( cos frac{pi}{7}+cos frac{2 pi}{7}+cos frac{3 pi}{7}+cos frac{4 pi}{7}+ )
( cos frac{5 pi}{7}+cos frac{6 pi}{7}+cos frac{7 pi}{7}= )
( A )
в.
c.
D.
11
545 27. For x e(0,), the equation sinx +2sin 2x -sin 3x = 3 has
(JEE Adv. 2014)
(a) infinitely many solutions
(b) three solutions
(c) one solution
(d) no solution
11
546 15.
Find the smallest positive number p for which the equation
cos( sin x) = sin(pcos x) has a solution x € (0.27)
(1995 – 5 Marks)
11
547 Illustration 4.65
Solve cos 2x > sinxl, xe
Bla
11
548 Illustration 4.1
Solve the equation sinx + cosx = 1.
11
549 ( frac{1}{sin ^{2} theta}-cot ^{2} theta ) is equal to
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 2 )
D. -2
11
550 If ( sin theta=-0.6, ) the find the quadrant
from which the terminal arm making an
angle of ( theta^{circ} ) passes.
A . I quadrant
B. II quadrant
c. III quadrant
D. IV quadrant
11
551 3. tan 100° + tan 125° + tan 100° tan 125° is equal to
b. 1/2
c. -1
d. 1
a. 0
11
552 If ( cos A=frac{sqrt{3}}{2}, ) then ( tan 3 A= )
( mathbf{A} cdot mathbf{0} )
B. ( frac{1}{2} )
c. 1
( D cdot alpha )
11
553 If ( tan ^{2} alpha=1-p^{2}, ) then ( sec alpha+tan ^{3} alpha )
( operatorname{cosec} alpha= )
11
554 59. The general solution of cos x cos 6x = -1 is
a. x = (2n + 1) n, ne Z b. x = 2nt,n e Z
c. x = nt, n e Z d. none of these
d.
11
555 75. The number of ordered pairs which satisfy the equation
x + 2x sin(xy) + 1 = 0 are (where y e [0, 21])
a. 1
b. 2
c. 3
d. O
11
556 2. Iff (x) = 2(7 cos x + 24 sin x)(7 sinx – 24 cos x), for every
xe R, then maximum value of (f(x))/4 is
11
557 Find the range of ( 3 cos theta-4 sin theta ) 11
558 – VI
+ sin
+
5. In triangle ABC, prove that sin
Hence, deduce that
2
T + C
1
T + A
COS 4
+ B
COS 4
COS4
=;
11
559 Illustration 2.48 Find the range of
f(x) = sin’x – 6sin x + 9+3.
11
560 ( A ) vertical tower stands on a declivity
which isinclined at ( 15^{0} ) to the horizon.
From the foot of the tower a man
ascends the declivity for80 feet and then finds that the tower subtends an
angle of ( 30^{0} . ) The height of the tower is
A ( cdot 20(sqrt{6}-sqrt{2}) f t )
B . ( 40(sqrt{6}-sqrt{2}) f t )
c. ( 40(sqrt{6}+sqrt{2}) f t )
D. ( 40 mathrm{ft} )
11
561 If in a triangle ( A B C, sin A cos B=1 / 4 )
and ( 3 tan A=tan B, ) then the triangle
is
A. right angled
B. equilateral
c. isosceles
D. none of these
11
562 Find the slope of the inclination of the
line of the following:
( boldsymbol{theta}=mathbf{3 0}^{circ} )
A. ( sqrt{3} )
B. ( frac{1}{sqrt{3}} )
c. ( frac{2}{sqrt{3}} )
D. ( frac{sqrt{3}}{2} )
11
563 Arrange the following equations in decreasing order of their number of solutions in ( [mathbf{0}, mathbf{2} boldsymbol{pi}] )
I) ( 3 sin ^{2} theta+4 cos ^{2} theta=5 )
II) ( 4 sin ^{2} theta+3 cos ^{2} theta=frac{7}{2} )
|||| ( 3 mid sin ^{2} theta+4 cos ^{2} theta=4 )
A . ॥,|॥,
B. ।, ॥।, III
c. ॥ा,।, ॥
( mathbf{D} cdot||, ) И,
11
564 Find ( x ) from the figure 11
565 ( A=cos ^{2} theta+sin ^{4} theta, ) find range of ( A ) 11
566 Prove that ( tan left(2 times 30^{circ}right)=frac{2 tan 30^{circ}}{1-tan ^{2} 30^{circ}} ) 11
567 The angle subtended at the centre of
circle of radius 3 metres by an arc of length 1 metre is equal to
A ( cdot 20^{circ} )
B . ( 60^{circ} )
c. ( frac{1}{3} ) radian
D. 3 radian
11
568 5. Number of solutions of equation 2 sin cos? x–2sin
sin? x = cos? x – sin? x for x € [0, 41] is
a. 6
b. 8
d. 12
c. 10
11
569 10. Let A, B, C be three angles such that A = 7/4 and tan B
tan C = p. Find all possible values of p such that A, B, C
are the angles of a triangle.
11
570 If ( tan 9^{circ}=frac{P}{q}, ) then the value of
( frac{sec ^{2} 81^{circ}}{1+cot ^{2} 81^{circ}} ? )
11
571 12. If tan B =
nsin a cos a
, prove that tan(a – b) = (1 – n)
tan a
1- nsin’ a’
11
572 Convert ( frac{3 pi}{7} ) in degrees. 11
573 30. If cos? x +
(1 + tan? 2y) (3 + sin 3z) = 4, then
cos- x)
a. x is an integral multiple of a
b. x cannot be an even multiple of a
c. z is an integral multiple of a
d. y is an integral multiple of /2
11
574 2. Given A = sin²0+ costo, then for all real 0,
a. 1SAS2
b. 3/4 SAS1
c. 13/16 S AS1 d. 3/4 S A S 13/16
(UTLIFE 1080
11
575 The value / values of ( x ) is/are
( mathbf{A} cdot pm 5 sqrt{5} )
B. ( pm sqrt{5} )
( ^{mathrm{c}}+frac{1}{sqrt{5}} )
D. none of these
11
576 Illustration 2.3. Prove that
sec A-tan A
cOS A
COS A
sec A + tan A
11
577 Illustration 2.57 Prove that
cos(90° + ) sec(-o) tan(180°-O)
-=
sec(360°-) sin(180°+2) cot(90°-0)
-1.
11
578 17. In a triangle ABC, if A – B = 120° and sin –
, then the value of 8 cos C is
11
579 37. The number of solutions of sec²0+ cosec?0+ 2 cosec?e
= 8,0 S OS Tt/2 is
a. 4
b. 3
c. O
d. 2
11
580 Find all the values of ( theta ) which satisfy the equation: ( cos theta cdot cos 2 theta cdot cos 3 theta=1 / 4 ) 11
581 If ( 270^{circ}<A<360^{circ}, 90^{circ}<B< )
( 180^{circ}, cos A=frac{5}{13}, tan B=-frac{15}{8}, ) then
( sin (boldsymbol{A}+boldsymbol{B})= )
A ( cdot frac{140}{221} )
в. ( frac{171}{221} )
c. ( frac{140}{171} )
D. ( frac{221}{171} )
11
582 6. Let sin(@-a)
a
sin(0-3) b
a. cos (a-B)
c. sin (a + B)
cos(-a) C ac+bd
then
cos(0-B) d ad + bc
b. sin (a-B)
d. none of these
11
583 The number of solutions of the equation ( x^{3}+2 x^{2}+5 x+2 cos x=0 ) in ( [0,2 pi] ) is:
A .
в.
( c cdot 2 )
D. 3
11
584 When ( n ) is a even natural number then
the value of ( x ) is
This question has multiple correct options
A. 0
в. ( pi / 2 )
( mathrm{c} cdot 2 pi / 3 )
D. ( pi )
11
585 10.
The
portion of a vertical pole subtends an
angle tan — at a point in the horizontal plane through its
foot and at a distance 40 m from the foot. A possible height
of the vertical pole is
[2003]
(a) 80m (b) 20 m (c) 40m (d) 60 m.
iges un and
11
586 18. If sec x + sec? x =1 then the value of tan® x – tanº x –
2 tan- x + 1 will be equal to
a. 0
b. 1
c. 2
d. 3
11
587 72. The number of solutions of the equation
12 sin x-5312cos -scOS = 1 in [0, 1] is
a. 2 to contatto b.
3
c. 4 O
d. 5
0
0
11
588 Prove that: ( sin left(frac{pi}{2}-thetaright)=cos theta ) 11
589 1. If sin = = and cos O=
=- V, then the general value of
O is (n e Z)
2
a. 2n +
b. 2nt +
BOB
c. 2nt +
d. 2nt +
11
590 11. The number of solutions of the pair of equations
2 sin? 0 – cos 20= 0
2 cos0 – 3 sin 0 = 0
in the interval [0, 27] is
b. 1
c. 2
d. 4
(IIT-JEE 2007
a. 0
11
591 prove that:
( frac{sin 5 x-2 sin 3 x+sin x}{cos 5 x-cos x}=tan x )
11
592 Show that sin?5° + sinº 10° + sin? 15° + …
Illustration 2.60
+ sin 90º = 9-.
11
593 67. The total number of solutions of sin {x} = cos {x} (where
{} denotes the fractional part) in [0, 2T] is equal to
a. 5
b.
61
c. 8
d. none of these
11
594 Prove that ( 4left[sin ^{4} 30^{circ}+cos ^{4} 60^{circ}right]- )
( 3left[cos ^{2} 45^{circ}-sin ^{2} 90^{circ}right]=2 )
11
595 The value of ( 144^{circ} ) in circular measure is
A ( cdot frac{3 pi^{c}}{4} )
B. ( frac{2 pi^{c}}{3} )
c. ( frac{4 pi^{c}}{5} )
D. ( frac{5 pi^{c}}{6} )
11
596 13. Show that
sin x sin 3x
cos 3x cos 9x
cos 9x
cos 27x
[tan 27x – tan x].
11
597 Illustration 2.30 Evaluate the sine, cosine, and tangent of
each of the following angles without using a calculator:
300°, -4050 71 117
11
598 9. If x, y = [0, 21] and sin x + sin y = 2, then the value of
x + y is
a. T
b. 7/2
c. 310
d. none of these
11
599 Evaluate the value of ( sin frac{pi}{6}+cos frac{pi}{6} ) 11
600 10. Solve sinx+ sin ( 3 V1 –cos 2x)’ +sin?2x) = 0, 11
601 Illustration 2.5 Convert it/6 rad to degrees. 11
602 Let ( boldsymbol{f}(boldsymbol{x})=log left(log _{frac{1}{3}}left(log _{7}(sin boldsymbol{x}+boldsymbol{a})right)right) )
be defined for every real value of ( x, ) then
the possible value of ( a ) is This question has multiple correct options
( A cdot 7 )
B. 8
( c .9 )
D. 6
11
603 27. Let f(x) = cos(a, + x) + -cos(az + x)+ 2 cos(az + x)
+…+ – cos (an + x) where aj, az, … ane R. If f(x)
= f(x) = 0, then (x2 – x; may be equal to
a.
T
b
.
211
c. 30
d.
1/2
11
604 If ( 7 cos ^{2} theta+3 sin ^{2} theta=4 ) and ( theta ) is in first
quadrant .Show that ( boldsymbol{theta}=mathbf{6 0}^{circ} )
11
605 If ( boldsymbol{A}=left{boldsymbol{x} epsilon boldsymbol{R} / frac{pi}{4} leq boldsymbol{x} leq frac{pi}{3}right} quad ) and
( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x}-boldsymbol{x} quad, ) then ( boldsymbol{f}(boldsymbol{A})= )
A ( cdotleft[frac{sqrt{3}}{2}-frac{pi}{3}, frac{1}{sqrt{2}}-frac{pi}{4}right] )
B. ( left[-frac{1}{sqrt{2}}-frac{pi}{4}, frac{sqrt{3}}{2}-frac{pi}{3}right] )
( mathbf{c} cdotleft[-frac{pi}{3}, frac{-pi}{4}right] )
D. ( left[begin{array}{c}pi \ hline 4end{array}, frac{pi}{3}right. )
11
606 ( frac{sin theta+sin 2 theta}{1+cos theta+cos 2 theta}=tan theta ) 11
607 The value of
( cot 15^{circ} cot 20^{circ} cot 70^{circ} cot 75^{circ} ) is equal
to
A . -1
B. 0
( c .1 )
D. 2
11
608 Assertion
The equation ( a sin x+cos 2 x=2 a-7 )
possesses a solution if ( boldsymbol{a} in[mathbf{2}, mathbf{6}] )
Reason
( -1 leq sin x leq 1 forall x in R )
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is
not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
11
609 ( sec A=frac{12}{5} ) for some value of angle ( A )
A. True
B. False
11
610 Find the value of ( csc 10^{circ}-sqrt{3} sec 10^{circ} ) 11
611 16. If tan
*
+
+
prove that sin y – 3+sin²x
sin x
1+ 3 sin- x
11
612 Convert ( frac{5 pi}{6} ) in to radians. 11
613 Match the following 11
614 4. The equation (cosp – 1)x2 + (cosp)x + sinp = 0 in the
variable x has real roots. Then p can take any value in the
interval
a. (0,21)
b. (- 1,0)
d. (0, 1)
(IIT-JEE 1990)
11
615 ( int tan ^{4} x sec ^{2} x d x ) 11
616 Which of the following is correct (where ( boldsymbol{n} boldsymbol{epsilon} boldsymbol{N}) ) 11
617 Prove that ( cos 40^{circ}+cos 50^{circ}+ )
( cos 70^{circ}+cos 80^{circ}=cos 20^{circ}+cos 10^{circ} )
11
618 If ( alpha, beta ) are solutions of ( sin ^{2} x+ ) ( a sin x+b=0 ) and ( cos ^{2} x+c cos x+ )
( boldsymbol{d}=mathbf{0}, ) then ( sin (boldsymbol{alpha}+boldsymbol{beta}) ) equals
A ( cdot frac{2 a c}{a^{2}+c^{2}} )
B. ( frac{a^{2}+c^{2}}{2 a c} )
c. ( frac{2 b d}{b^{2}+d^{2}} )
D. ( frac{b^{2}+d^{2}}{2 b d} )
11
619 If ( frac{sin A}{sin B}=p ) and ( frac{cos A}{cos B}=q, ) find ( tan A )
and ( tan B )
11
620 Illustration 3.7
Illustration 3.7 If sin(A – B) – Tro.cos(A+B)- 20 find
If sin(A-B) =
find
the value of tan 2 A where A and B lie between 0 and /4.
11
621 13. If 0 SX S27, then 2cosecºx – y – y+1 s 12
a. is satisfied by exactly one value of y
b. is satisfied by exactly two value of x
c. is satisfied by x for which cosx = 0
Cod. is satisfied by x for which sinx = 0
11
622 Find the value of ( cos 75^{circ} ) 11
623 2. The general solution of the trigonometric equation
sinx + cos x = 1 is given by
a. x = 2nt, n= 0, #1, #2, …
b. x = 2n1 + Tt/2; n= 0, +1, +2, …
c. x = nt+ (-1)” ?-?n=0, #1, #2, …
4 4 4
d. none of these
(IIT-JEE 1981)
11
624 then prove that
Illustration 3.10 If a, b, ye 0,-
sin(a+B+y)
sin a +sin ß + sin y
-<1.
11
625 w
1. The expression sin“ (3x –a)+ sin“Gr + c
-2 sin* (3x+a)=sin(57 –is equal to
a. 0
c. 3
b. 1
d. none of these
11
626 Illustration 3.36
In quadrilateral ABCD, if
GOS
COS
then find the value of sin
sin — sin — sin
la
11
627 Find the maximum value of ( sqrt{3} sin x+ )
( cos x )
11
628 10. Number of roots of the equation
SUN
NIS
tan
x-
12
4
) 2(0.25) cos 2x + 1 = 0 is
11
629 TTC
16.
Find all values of in the interval
satisfying the
2
equation (1 – tan 0) (1 + tan o) sec2
+ 2 tan e=0.
(1996 – 2 Marks)
11
630 llustration 4.6
Solve 16sinx + 16c0s? x = 10,0 5x< 27.
11
631 + 2 cot
771
44. The value of cot-
16
a. 4
c. -2
157
+ cot –
16
51007
b. 2
d. -4
11
632 A pole of height 20 ft has a shadow of length ( 11.55 mathrm{ft} ) at a particular instant of
time. Find the angle of elevation (in degree) of the sun at this point of time.
( mathbf{A} cdot 90 )
B. 60
( c cdot 30^{circ} )
D. 45
11
633 COS
Illustration 3.38 In AABC, sin C + cos C + sin(2B + C) –
cos (2B + C) = 2V2. Prove that AABC is right-angled
isosceles.
11
634 The value of ( cos left(-1044^{circ}right) ) is ( frac{(sqrt{5}+1)}{4} )
A . True
B. False
11
635 96. 180sxs then range of fw) = sec (*_*)+ see(++)
sec
11
636 Simplify, using trigonometric tables ( sin 50^{circ} 26^{prime}+cos 18^{circ}+tan 70^{circ} 12^{prime} ) 11
637 19. If cotx = cot(x – y) cot(x – z), then cot2x is equal to
(where x #T /4)
a.
(tan y + tan z)
b. =(cot y+cot z)
.
.
.
– (sin
(sin y+sin z)
d. none of these
C
D
11
638 ( ln a Delta A B C, cos B cdot cos C+ )
( sin B cdot sin C cdot sin ^{2} A=1 . ) Then the
triangle is
A. right – angled isosceles
B. isosceles whose equal angles are greater then ( pi / 4 )
c. equilateral
D. none of these
11
639 7. Show that 16008 ()cos ( 15 cos 15 cos 1657) – 1. 11
640 The maximum valuue of the expression ( frac{1}{sin ^{2} theta+3 sin theta cos theta+5 cos ^{2} theta} ) 11
641 If ( sin ^{-1} x=y, ) then
A. ( 0 leq y leq pi )
B. ( -frac{pi}{2} leq y leq frac{pi}{2} )
c. ( 0<y<pi )
D. ( -frac{pi}{2}<y<frac{pi}{2} )
11
642 Express the following angle in terms of first-quadrant reference angle:
( tan 336^{circ} )
A . ( tan 45 )
B . ( -tan 36 )
c. ( -tan 24 )
D. ( tan 24 )
11
643 ( a=operatorname{cosec} 2^{0}, b=operatorname{cosec} 2, ) then which of
the following holds true?
( mathbf{A} cdot a=b )
В ( . a>b )
c. ( a<b )
D. ( a=2 b )
11
644 If ( tan theta=frac{sin alpha-cos alpha}{sin alpha+cos alpha}, ) then
( A cdot sin alpha-cos alpha=pm sqrt{2} sin theta )
B. ( sin alpha+cos alpha=pm sqrt{2} cos theta )
( mathbf{c} cdot cos 2 theta=sin 2 alpha )
( D cdot sin 2 alpha+cos 2 alpha=0 )
11
645 Find the following:
( sin left[frac{pi}{2}-sin ^{-1}left(frac{-sqrt{3}}{2}right)right] )
11
646 and
Illustration 3.82 In triangle ABC, if cot A. cot C = =
cot B.cot C = =, then the value of tan C is
11
647 ( ln Delta A B C, ) a ( sin (B-C)+b sin (C- )
( A)+c sin (A-B)= )
( A )
B. ( a+b+c )
c. ( a^{2}+b^{2}+c^{2} )
D ( cdot 2left(a^{2}+b^{2}+c^{2}right) )
11
648 The number of solutions of ( sin 3 x= )
( cos 2 x, ) in the interval ( left(frac{pi}{2}, piright) ) is
( A cdot 3 )
B. 4
( c cdot 2 )
D.
11
649 If ( sin A+cos A=sqrt{2} ) then ( sin A )
( cos A ) is equal to:
11
650 Find the value of ( operatorname{cosec}^{2} 45^{circ}-cot ^{2} 45^{circ} ) 11
651 If ( operatorname{Sin}^{2} A+operatorname{Cos}^{2} B+operatorname{Sin}^{2} C=1, ) then
the triangle ( A B C ) is
A. isosceles
B. equilateral
c. right angled
D. scalene
11
652 Illustration 3.76 Prove that in triangle ABC, cos?A + cos²B
– cos C = 1 – 2 sin A sin B cos C.
11
653 If ( cos ^{2} x=cos ^{4} x ) then find the
minimum and maximum value of
( sin ^{2} x+cos ^{4} x, ) for all real values of ( theta )
11
654 Degree measure of ( left(frac{7 pi}{6}right)^{c} ) is
A ( .210^{circ} )
B . ( 240^{circ} )
( c cdot 270 )
D. None of these
11
655 3. IfK=sin(īt/18) sin(57/18) sin(77/18), then the numerical
value of K is
(IIT-JEE 1993)
11
656 13. The value of f(a) Vcosec’a – 2 cot a
+Vcoseca – 2 cot a can be
a. 2 cot a
b. – 2 cot a
c. 2
b. -2
11
657 Illustration 3.30
Prove that cos 18º – sin 18° = 12 sin 27°
11
658 ( frac{cos ^{2} 30^{circ}+cos 30^{circ} sin 30^{circ}+sin ^{2} 30^{circ}}{cos ^{3} 30^{circ}-sin ^{3} 30^{circ}} ) 11
659 The value of ( 1+cot ^{2} A ) is
( mathbf{A} cdot cos ^{2} A )
B. ( sec ^{2} A )
( mathbf{c} cdot tan ^{2} A )
D. ( operatorname{cosec}^{2} A )
11
660 ( sin ^{3} x-sin ^{3}left(240^{0}-xright)+sin ^{3}left(240^{0}+right. )
( boldsymbol{x}) in )
( [-boldsymbol{k}, boldsymbol{k}]=rangle boldsymbol{k}= )
( A )
B. ( 1 / 4 )
( c cdot 3 / 4 )
( D cdot 5 / 4 )
11
661 Illustration 2.61 Ifsin(120°-a)=sin(120°– B), 0<a, ß<TT,
then find the relation between a and B.
11
662 If ( tan A+sec A=3, ) Find the value of ( sin )
( mathbf{A} )
11
663 ( frac{sin (boldsymbol{A}-boldsymbol{B})}{cos boldsymbol{A} cos boldsymbol{B}}+frac{sin (boldsymbol{B}-boldsymbol{C})}{cos boldsymbol{B} cos boldsymbol{C}}+ )
( frac{sin (boldsymbol{C}-boldsymbol{A})}{cos boldsymbol{C} cos boldsymbol{A}}=boldsymbol{0} )
11
664 17. If 5(tan?x-cos2x)=2cos 2x +9, then the value of cos 4x is:
[JEEM 2017
11 L
lations of the equati
11
665 8. If cos 4x = a + a,cos²x + a cosx is true for all
values of x e R, then the value of 5a, + aj + az
11
666 22.
Given both O and o are acute angles and sin
=
cos o = =, then the value of 0 + 0 belongs to
(2004)
11
667 60. The number of solutions the equation cos(O). cos(110) = 1
has
b. 2 O
c. 1
d. infinite
a. 0
11
668 Find general solution for ( x ) ( sin 8 x-cos 6 x=sqrt{3}(sin 6 x+cos 8 x) ) 11
669 Illustration 4.55
Solve cos 40+ sin 50= 2.
11
670 12. The value of tan?
tan? 24 tan2 341 is
a. – 3
c. -5
b. 7
d. none of these
11
671 What is the greatest value of ( theta ) lying between ( 0^{0} ) and ( 720^{0} ) whose tangent is ( -frac{1}{sqrt{3}} ? )
A ( cdot 690^{circ} )
( ^{0} )
в. ( 510^{circ} )
c. ( 330^{circ} )
D. ( 150^{circ} )
11
672 If ( cos A=frac{4}{5}, cos B=frac{12}{13}, frac{3 pi}{2}<A, B< )
( 2 pi, ) find the values of the following.
( (i) cos (A+B) )
(ii) ( sin (A-B) )
11
673 The value of ( sec ^{2} A tan ^{2} B- )
( tan ^{2} A sec ^{2} B ) is
( mathbf{A} cdot tan ^{2} B-tan ^{2} A )
B ( cdot sec ^{2} B+sec ^{2} A )
c. ( tan ^{2} B-sec ^{2} A )
( mathbf{D} cdot sec ^{2} B-tan ^{2} A )
11
674 A wheel makes 12 revolutions per hour
The radians it turns through in 20
minutes is:
( A cdot 8 pi^{c} )
В . ( 16 pi^{c} )
( c cdot 24 pi^{c} )
D. ( 32 pi^{c} )
11
675 If ( tan A=frac{1}{2} ) and ( tan B=frac{1}{3}, ) then the
value of ( boldsymbol{A}+boldsymbol{B} ) is
A ( cdot frac{pi}{6} )
в. ( pi )
c. zero
D.
11
676 Solve: ( cos ^{2} x-2 cos x=4 sin x- )
( sin 2 x, 0 leq x leq pi )
11
677 14. Determine the smallest positive value of x (in degrees) for
which
tan(x +100°)=tan(x +50°) tan(x) tan (x – 50°).
(1993 – 5 Marks)
11
678 If ( tan (A-B)=1, sec (A+B)=frac{2}{sqrt{3}} )
then the smallest +ve value of B is?
A ( cdot frac{25 pi}{24} )
в. ( frac{19 pi}{24} )
c. ( frac{13 pi}{24} )
D. ( frac{11 pi}{24} )
11
679 What is the reference angle, ( rho ? )
( mathbf{A} cdot rho=30^{circ} )
( mathbf{B} cdot rho=45^{circ} )
( mathbf{C} cdot rho=56^{circ} )
D ( cdot rho=270^{circ} )
11
680 2. The number of values of 0 in the interval
(
NT
such that ex mat for n= 0, +1, + 2 and
such that 0 + – for n = 0,
I 22
tan O=cot 50 as well as sin 20 = cos 40 is
1,
2 and
TOO
11
681 Illustration 4.42 Solve 4 cot20= cote- tane. 11
682 Illustration 4.44
Solve 73 cos 0+ sin 0 = V2.
11
683 Illustration 4.51 If 3 sinx + 4 cos ax = 7 has at least one
solution, then find the possible values of a.
11
684 If ( mathbf{0} leq boldsymbol{x} leq mathbf{3} boldsymbol{pi}, mathbf{0} leq boldsymbol{y} leq )
( 3 pi ) and ( cos x . sin y=1 ) then the
possible number of values of the
ordered pair ( (x, y) ) is
( A cdot 6 )
B. 12
c. 8
D. 15
11
685 6.
If

X
tan ay
, then show that
tan(6+) tan(8+B) tan(6+y)’
2 * sin?(a – B)=0.
x – y
11
686 Which of the following do/does not
reduce to unity?
This question has multiple correct options
A ( frac{sin left(180^{0}+Aright)}{tan left(180^{0}+Aright)} frac{cot left(90^{0}+Aright)}{tan left(90^{0}+Aright)} frac{cos left(360^{0}-Aright) operatorname{cosec} A}{sin (-A)} )
B. ( frac{sin (-A)}{sin left(180^{0}+Aright)}-frac{tan left(90^{0}+Aright)}{cot A}+frac{cos A}{sin left(90^{0}+Aright)} )
C. ( frac{sin 24^{0} cos 6^{0}-sin 6^{0} cos 24^{0}}{sin 51^{0} cos 69^{0}-cos 51^{0} sin 69^{0}} )
D. ( frac{cos left(90^{0}+Aright) sec (-A) tan left(180^{0}-Aright)}{sec left(360^{0}+Aright) sin left(180^{0}+Aright) cot left(90^{0}-Aright)} )
11
687 Which of the following is correct:
( A cdot sin 1^{circ}>sin 1 )
B. ( sin 1^{circ}<sin 1 )
( mathbf{c} cdot cos 1^{circ}<sin 1 )
D. None of these
11
688 The value of ( sin 105^{0} ) is
A ( cdot frac{sqrt{3}-1}{2 sqrt{2}-1} )
B. ( frac{sqrt{3}-1}{sqrt{2}-1} )
c. ( frac{sqrt{3}+1}{2 sqrt{2}} )
D. ( frac{sqrt{3}+1}{sqrt{2}} )
11
689 Illustration 2.56 Prove that sin(-420°) (cos 390°) +
cos (-660°) (sin 330°) = -1.
11
690 12. If cos(x + 7/3) + cos x = a has real solutions, then
a. number of integral values of a are 3
b. sum of number of integral values of a is 0
c. when a = 1, number of solutions for xe [0, 276] are
3
d. when a = 1, number of solutions for xe [0, 2T) are
11
691 Prove that
( frac{cos e c A+1}{cos e c A-1}=frac{1+sin A}{1-sin A} )
11
692 Prove that:
( sin ^{6} theta+cos ^{6} theta+3 sin ^{2} theta cos ^{2} theta=1 )
11
693 If ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi}, ) then prove the
following ( sin 2 A+sin 2 B+sin 2 C= )
( 4 sin A cdot sin B cdot sin C )
11
694 sin- – cos3
Solve –
2 + sin x
2
COS X
Illustration 4.20
11
695 Find the value of ( boldsymbol{theta}, ) if
( cos theta=0.9664 )
11
696 Illustration 4.43 Find the most general solution of
21+ |cos x + cos x + |cosxp + … =4
11
697 21. Let S = {0€[-21, 21t]: 2 cos 0+3 sino=0}.
Then the sum of the elements of S is:
[JEEM 2019-9 April (M
11
698 12.
If exp {(sinx + sin^x + sinºx + …………… (0) In 27
satisfies the equation x2- 9x + 8 = 0, find the value of
| cosx 0<x< ".
(1991 – 4 Marks)
cos x + sinx
com <**
29. septes
11
699 Given ( boldsymbol{A}=sin ^{2} boldsymbol{theta}+cos ^{4} boldsymbol{theta}, ) then for all
real ( theta, ) which of the following is true?
A. ( 1 leq A leq 2 )
в. ( frac{3}{4} leq A leq 1 )
c. ( frac{13}{16} leq A leq 1 )
D. ( frac{3}{4} leq A leq frac{13}{16} )
11
700 Find the value of ( x ) if ( sin ^{-1}left(frac{2}{3}right)+ ) ( sin ^{-1}left(frac{2}{3}right)=sin ^{-1} x ) 11
701 if ( sin theta=frac{1}{2} ) and ( cos theta=-frac{sqrt{3}}{2}, ) then
the general value of ( theta i s(n epsilon z) )
A ( cdot 2 n pi+frac{5 pi}{6} )
В ( cdot 2 n pi+frac{pi}{6} )
c. ( _{2 n pi+frac{7 pi}{6}} )
D. ( 2 n pi+frac{pi}{4} )
11
702 72. The value of tan 9° – tan 27° – tan 63° + tan 81° is
a. 2
b. 3
c. 4
d. none of these
11
703 In a triangle ( A B C, ) the least value of ( tan ^{2} frac{A}{2}+tan ^{2} frac{B}{2}+tan ^{2} frac{C}{2} ) is
( mathbf{A} cdot mathbf{1} )
B.
( c cdot sqrt{3} )
D. ( frac{2}{3} )
11
704 7
63. If cos 3x + sin 2x –
– 2, then x is equal to (ke Z)
6
a. (6x +1)
ok +1
b. (k-1)
(2k +1)
d. none of these
11
705 The value of ( sec ^{3} theta, ) where ( theta ) is the acute
angle between the plane faces of a regular tetrahedron, is
11
706 ( ln Delta A B C, ) if ( 2 sin ^{2} B+2 sin ^{2} C+ )
( cos 2 A=1 ) then the triangle is
A. right angled
B. isosceles
c. equilateral
D. right angled and isosceles
11
707 ( sin frac{7 pi}{4}=frac{1}{sqrt{2}} )
A. True
B. False
11
708 , where x e 1st quadrant, then tan-
62. If sin x + cos x =
is equal to
2
7-2
d. none of these
11
709 ( ln a Delta A B C, ) prove that
( (a) cos (A+B)+cos C=0 )
(b) ( tan frac{(A+B)}{2}=cot frac{C}{2} )
11
710 ( sin 20^{0} times cos 40^{0} times tan 45^{0} times cos 90^{0}= ) 11
711 6. Number of solutions of the equation 4(cos22 x + cos2x+1)
+ tan x(tan x-2/3) = 0) in [0, 21] is
a. 0
b. 1
c. 2
d. 3
11
712 28. If 0 < x < 21, then the number of solutions of 3(sin x +
cosx) – 2(sinx + cosx) = 8 is
a. 0
b. 1
c. 2
due to di
11
713 Illustration 2.49 If (x, y) satisfies the equation 1 + 4x – x2
= 19 sec? y + 4 cos ec?y, then find the value of x and tan”y.
11
714 coto
Sa
16. Let f(0) =
16. Let f(0) = 1 + cote and a + ß
, then the value of
4 , then ti
f(a) f(B) is
b.

2
c. 2
d. none of these
11
715 The possible value(s) of ‘ ( boldsymbol{theta} ) ‘ satisfying
the equation ( sin ^{2} theta tan theta+ )
( cos ^{2} theta cot theta-sin 2 theta=1+tan theta+ )
( cot theta, ) where ( theta in[0, pi] ) is/are
A.
в. ( pi )
c. ( frac{7 pi}{12} )
D. None of these
11
716 26. The least value of 2 sin’e + 3 coso is
a. 1
b. 2
c. 3
d. 5
11
717 Illustration 3.27 If sin?(0 – a) cos a = cos(-a)sin a =
m sin a cos a, then prove that m| 3 –
11
718 Solve
( tan theta=frac{sin alpha-cos alpha}{sin alpha+cos alpha} )
11
719 In triangle ( A B C, ) right-angled at ( B ), if ( tan A=frac{1}{sqrt{3}} ) find the value of:
(i) ( sin A cos C+cos A sin C )
11
720 85. The equation tan*x – 2 sec²x + a=0 will have at least one
solution if
a. 1<a s4
b. a 2
c. a <3 O
d
d. none of these
none
11
721 If ( sqrt{3} tan theta=3 sin theta, ) find the value of
( sin ^{2} theta-cos ^{2} theta )
11
722 6. Convert 18 degree into radians.
rad
rad
11
723 Find the radian measure corresponding
to the degree ( -47^{circ} 30^{prime} )
A ( cdot frac{-19 pi}{72} r a d )
В ( cdot frac{19 pi}{72} r a d )
c. ( frac{13 pi}{72} ) rad
D. None of these
11
724 The value of ( cos ^{2} 45^{circ}-sin ^{2} 15^{circ} ) is
( A cdot frac{3}{4} )
B. ( frac{sqrt{5}}{4} )
( c cdot frac{sqrt{3}}{4} )
D. ( frac{sqrt{5}+1}{4} )
11
725 Express in degree form ( left(frac{5 pi}{12}right)^{c} )
A ( .60^{circ} )
B . 45
( c cdot 180 )
D. ( 90^{circ} )
11
726 The number of solutions of the equation ( mathbf{1}+sin boldsymbol{x} cdot sin ^{2} frac{boldsymbol{x}}{mathbf{2}}=mathbf{0}[-boldsymbol{pi}, boldsymbol{pi}] ) is
A . zero
B.
( c cdot 2 )
( D )
11
727 Convert ( 40^{circ} 20^{prime} ) into radian measure.
A ( cdot frac{121}{540} pi ) radians
в. ( frac{121}{570} pi ) radians
c. ( frac{120}{513} pi ) radians
D. None
11
728 is
1. If 0 Sost and 81sin’e + 81c0s? 0 = 30, then
a. 30°
b. 60°
c. 120°
d. 150°
11
729 Illustration 4.27 Solve 5 cos 20 + 2 cos2 – +1 = 0,
– 1<<.
11
730 If ( sin 25^{circ} cdot sin 35^{circ} cdot sin 85^{circ}=frac{cos x^{circ}}{a} ) wher
argument of ( cos ) is acute and positive then find the value of ( (x+a) )
11
731 What will be the values of ( theta ) between ( 0^{circ} )
and ( 360^{circ} ) if ( sin theta=-frac{sqrt{3}}{2} )
A ( cdot 30^{circ}, 330^{circ} )
В. ( 60^{circ}, 120^{circ} )
c. ( 135^{circ}, 315^{circ} )
D . ( 240^{circ}, 300^{circ} )
11
732 If ( frac{x}{cos theta}=frac{y}{cos left(theta-frac{2 pi}{3}right)}= )
( frac{z}{cos left(theta+frac{2 pi}{3}right)}, ) then ( x+y+z ) is equal to
( A )
B.
( c cdot-1 )
D. none of these
11
733 Degree measure of ( left(frac{1}{4}right)^{c} ) is
A ( cdot 15^{circ} 19^{prime} 5^{prime prime} )
B . ( 14^{circ} 19^{prime} 5^{prime prime} )
( mathbf{c} cdot 15^{circ} 18^{circ} 6 ” )
D. ( 14^{circ} 18^{circ} 6 ” )
11
734 Find the value of ( left(4 cdot sin 30^{0} cos 30^{0}right)^{2} ) 11
735 The value of ( 60^{circ} ) in centesimal system is
A ( cdot frac{100^{9}}{3} )
в. ( frac{200^{9}}{3} )
c. ( frac{140^{9}}{3} )
D. ( frac{160^{9}}{3} )
11
736 Express 1.2 rad in nereast integer
degree measure.
11
737 1. If tan 0+ sin 0 = m and tan 0 – sin 0= n, then
a. m² – n² = 4mn b. m² + n² = 4mn
c. m – n2 = m² + 12 d. m – n2 = 47mn
11
738 ( sin theta=cos theta ) for all values of ( theta )
Enter 1 for true and 0 for false
11
739 In a triangle ( A B C )
( boldsymbol{b} cos (boldsymbol{C}+boldsymbol{theta})+boldsymbol{c} cos (boldsymbol{B}-boldsymbol{theta})= )
A ( . a cos theta )
B. ( a sin theta )
c. ( a tan theta )
D. a cot ( theta )
11
740 Given: ( cos A=0.6 ; ) find all other
trigonometrical ratios for angle ( boldsymbol{A} )
11
741 11.
For O<O<
the solution (s) of
m
Cosec
cosec
m
=1
is (are)
(2009)
11
742 Solve ( left(sec ^{2} theta-1right)left(1-operatorname{cosec}^{2} thetaright)=-1 ) 11
743 Radian measure of ( 175^{circ} 45^{prime} ) is
A ( cdot frac{700}{720} pi )
в. ( frac{703}{720} pi )
c. ( frac{705}{720} pi )
D. ( frac{710}{720} pi )
11
744 Solve the equation ( : cot x-2 sin 2 x=1 )
A ( cdot x=frac{pi}{8}+frac{K pi}{2} )
в. ( x=frac{pi}{4}+frac{K pi}{4} )
c. ( _{x}=frac{pi}{6}+frac{K pi}{3} )
D. ( x=frac{2 pi}{3}+frac{K pi}{3} )
11
745 ( frac{tan theta}{left(1+tan ^{2} thetaright)^{2}}+frac{cot theta}{left(1+cot ^{2} thetaright)^{2}}=sin theta )
( cos theta quad, ) then ( theta ) is
( mathbf{A} cdot 2 pi )
в. ( frac{pi}{56} )
( c cdot frac{pi}{2} )
D.
11
746 Ilustration 2.54
Prove that sin
<
< tan
for OE (0, 70/2).
11
747 ( pi^{c} )
( frac{pi}{5} ) in sexagesimal measure is
A ( cdot 18^{circ} )
B. ( 36^{circ} )
( c cdot 54^{circ} )
D. 72
11
748 Illustration 4.29
cos?x.
Solve the equation
sinx – cos x =
11
749 66. If cos 2x = 1, the value of cos x
is (0° S XS 45°)
(1) 2
(2) O
(3) 1
(4) 2
000
e) ond
11
750 It is known that ( sin beta=frac{4}{5} & 0<beta<pi )
then the value of
( frac{sqrt{mathbf{3}} sin (boldsymbol{alpha}+boldsymbol{beta})-frac{mathbf{2}}{mathbf{c o s}} frac{boldsymbol{pi}}{mathbf{6}} cos (boldsymbol{alpha}+boldsymbol{beta})}{sin boldsymbol{alpha}} )
A. independent of ( alpha ) for all ( beta ) in ( (0, pi) )
В. ( frac{5}{sqrt{3}} ) for ( tan beta0 )
( D . ) none
11
751 10. Let x = sin 1°, then the value of the expression
11
cos 0°.cos 1° cos1º.cos 2° cos 2º.cos 3°
1
xord)
-+-
cos 44º. cos 450 is equal to
a. x
c. √21x
b. 1/x
d. x/ 12
11
752 If ( frac{sin x}{1+sec x}+frac{sin x}{sec x-1}=2 ) where
( 0^{circ}<x<90^{circ} ) then cosec ( x ) has the value
equal to
( mathbf{A} cdot mathbf{1} )
B. 2
c. ( sqrt{2} )
D. ( sqrt{3} )
11
753 Show that ( cot 7 frac{1^{circ}}{2}=sqrt{2}+sqrt{3}+sqrt{4}+ )
( sqrt{6} )
11
754 Illustration 3.52 Find the least positive value of x satisfying
sin22x + 4 sin^x- 4 sinx cos²x 1
4-sin2x – 4 sin²x
11
755 Illustration 2.47 Find the range of f(x) = sin?x – 3 sinx +2 11
756 Prove that: ( frac{1}{operatorname{cosec} A-cot A}-frac{1}{sin A}= )
( frac{1}{sin A}-frac{1}{operatorname{cosec} A+cot A} )
11
757 If ( sin (A+B)=1 ) and ( cos (A-B)= )
( frac{sqrt{3}}{2} . ) then find the minimum positive
value of ( boldsymbol{A} ) and ( boldsymbol{B} ) is
A ( cdot 60^{circ}, 30^{circ} )
( ^{circ} 0^{circ} )
B . ( 75^{circ}, 15^{circ} )
( mathbf{c} cdot 45^{circ}, 60^{circ} )
D. ( 45^{circ}, 45^{circ} )
11
758 TT
Illustration 3.106 Let a, b, y> 0. and a +B+ y = 5. Then
prove that tan a tan B + ytan ſtan y + tan a tan y s 13
11
759 The domain of the function ( y= )
( sqrt{sin x+cos x}+sqrt{7 x-x^{2}-6} ) is
( left[p, frac{q pi}{4}right] cupleft[frac{r pi}{4}, sright] ) then value of ( p+ )
( boldsymbol{q}+boldsymbol{r}+boldsymbol{s} ) is
11
760 Simplify ( sqrt{2+sqrt{2+2 cos left(pi+60^{0}right)}} ) 11
761 (1994)
10. Let 0<x< then (sec2x – tan2x) equals
an
X-
11
762 Illustration 3.77 In triangle ABC, prove that
sin(B + C – A) + sin(C + A – B) + sin(A + B-C) = 4 sin A
sin B sin C.
11
763 Prove that,
( cos 20^{circ} cos 40^{circ} cos 60^{circ} cos 80^{circ}=frac{1}{16} )
11
764 The value of ( 45^{circ} ) in centesimal system
is
A ( cdot 25^{9} )
B. ( 50^{text {9 }} )
( mathrm{c} cdot 75^{9} )
D. ( 100^{9} )
11
765 If x2 + y2 = 4, then find the maximum value
Illustration 3.94
of x² + y²
xty
11
766 COS X
COS X
5. If cos(x – y), cosx and cos(x + y) are in H.P., then cos x
(IIT-JEE 1997)
Sec
11
767 Assertion
The minimum value of the expression
( sin alpha+sin beta+sin gamma ) where ( alpha, beta, gamma ) are
real number such that ( boldsymbol{alpha}+boldsymbol{beta}+boldsymbol{gamma}=boldsymbol{pi} )
is negative because-
Reason
( alpha, beta, gamma ) are angles of a triangle.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
768 If ( tan theta+sec theta=2 x ) then prove that
( sec theta=x+frac{1}{4 x} )
11
769 In which quadrant ( 196^{circ} ) lies?
B. quadrant III
c. quadrantı
D. quadrant IV
11
770 4. The real roots of the equation cos’x + sin x = 1 in
the interval (-1, T) are

–, and
11
771 One root of the equation ( cos x-x+ )
( frac{1}{2}=0 ) lies in the interval?
A ( cdotleft(0, frac{pi}{2}right) )
В. ( left(-frac{pi}{2}, 0right) )
c. ( left(-frac{pi}{2}, piright) )
D. ( left(pi, frac{3 pi}{2}right) )
11
772 22. One of the general solutions of 4 sin o sin 20 sin 40 =
sin 30 is
a. (3n +1) 6/12, Vne Z
b. (4n+1) 7/9, ne z
c. (3n+1) 7/9, ne z
d. (3n+1) 7/3, ne z
11
773 Sum of all the values of ( x ) in the interval
( [0,100 pi] ) satisfying the equation
( sin x=0 ) is
( mathbf{A} cdot 4950 pi )
B. ( 5050 pi )
c. ( 5151 pi )
D. none of these
11
774 Illustration 2.45
Find the range of f(x) =
5 sin x
6
11
775 10. If sec4 e + sec? 0 = 10 + tanº e + tan’ 6, then sin? 0 =
Alw
wilt WIN
d.
alu
11
776 Number of solations of the equation (1993 – I Mark)
wax* x2cos x lying in the interval [0,21] is:
R 0 (0) 1 (c) 2
(d) 3
11
777 19. sing
become on
79. sinx + cos x = y2 – y + a has no value of x for any value
of y if a belongs to
a. (0, 13) b. (-13,0)
c. (- 0o – √3) d. (√3,0)
11
778 ( cos ^{2} A+cos ^{2}left(120^{circ}+Aright)+ )
( cos ^{2}left(120^{circ}-Aright)= )
A ( cdot frac{3}{2} )
в. ( frac{1}{2} )
( c cdot frac{3}{18} )
D.
11
779 Illustration 3.102 In AABC, prove that
cosee 4 + cosec + cosce ©26.
cosec
+ cosec
— + cosec
NI
11
780 If ( sin alpha=frac{1}{2}, ) then find value of ( 3 sin alpha- )
( 4 sin ^{3} alpha )
11
781 Illustration 2.38
a. sin 3
c. sin 1
Which of the following is the least?
b. sin 2
d. sin 7
11
782 15. Let a and B be non-zero real numbers such that
2(cosß- cosa) + cosa cosß= 1. Then which of the following
is/are true?
(JEE Adv. 2017)
11
783 Prove the following identities:
( sin 3 x+sin 2 x-sin x= )
( 4 sin x cos frac{x}{2} cos frac{3 x}{2} )
11
784 19.
is equal to
2r sin a
1 + 2r cos a
a. tan²0
c. cot 20
b. cote
d. tan 20
11
785 Show that
( tan 36^{circ} tan 17^{circ} tan 54^{circ} tan 73^{circ}=1 )
11
786 87. If a, b e [0, 21) and the equation x2 +4 + 3 sin(ax + b) –
2x = 0 has at least one solution, then the value of (a + b)
can be
not.
b. Sy
d. none of these
11
787 30. The number of solutions of the equation cos 6x + tan²x +
cos 6x . tan²x = 1 in the interval [0, 21) is
b. 5
od = c.
6 20
d. 7
A
a. 4
11
788 ( cos frac{pi}{5} cos frac{2 pi}{5}=frac{1}{4} ) and ( cos frac{pi}{5}+ )
( cos frac{3 pi}{5}=frac{1}{2} ) without using standard
value
11
789 If ( tan theta+tan 2 theta+tan theta tan 2 theta=1 ) then
general value of ( boldsymbol{theta} ) is
( mathbf{A} cdot n pi ; n in I )
в. ( n pi pm frac{pi}{3} ; n in I )
c. ( frac{n pi}{3}+frac{pi}{12}, n in I )
D. None of these
11
790 If ( sec ^{2} theta=frac{4}{3}, ) then the general solution
of ( boldsymbol{theta} ) is
( mathbf{A} cdot 2 n pi pm pi / 6 )
в. ( 2 n pi pm pi / 3 )
c. ( n pi pm pi / 6 )
D. ( n pi pm pi / 3 )
11
791 1. The positive integer value of n> 3 satisfying the equation
+-
is
(IIT-JEE 2011)
( 377
sin
sin
27

n
)
sin/
In
11
792 14. Solve the equation sin’x cos 3x + cos’x sin 3x + 3 = 0. 11
793 71. If sin 0 + sinº0 = 1, then the val-
ue of cos 12 + 3 cos 10 m+ 3 cos8
0 + cos6 0 – 1 is
(1) 2
(2)

(3) 12
(4) O
11
794 If ( sin alpha+sin beta=frac{1}{2} ) and ( cos alpha+ )
( cos beta=frac{sqrt{3}}{2}, ) then value of ( 3 alpha+beta ) is
A ( .90^{circ} )
B . ( 0^{circ} )
( c cdot 120^{circ} )
D. ( 60^{circ} )
11
795 15. If cos 28° + sin 28° = k”, then cos 17° is equal to
the core 2 is equal to
k3
b. —
* Be
121 enero
V2
si
d. none of these
11
796 The value of ( sqrt{left(frac{1+cos theta}{1-cos theta}right)}-csc theta )
( cot theta= )
11
797 Match the statements of column ( I ) with
values of column ( boldsymbol{I} boldsymbol{I} )
11
798 88. The sum of all roots of sin ne log:( – )) = 0 in (0,27) is
2: 1313
a. 3/2
c. 9/2
b. 4
d. 13/3
11
799 Illustration 4.4 Solve
log(-x-6x)/10 (sin 3x+sin x) = log-x2-6x)/10 (sin 2x)
11
800 7. For the smallest positive values of x and y, the equation
2(sin x + siny) – 2 cos(x – y) = 3 has a solution, then
which of the following is/are true?
a. sin *+=
1
b
. cos
c. number of ordered pairs (x, y) is 2
d. number of ordered pairs (x, y) is 3
11
801 84. elsin xl + e sin xl + 4a = 0 will have exactly four different
solutions in [0, 21) if
a. a E R
b. a e
c. cele
c. ae –
d
d. none of these
. none of these
11
802 Find the cosine of ( 8^{circ} 12 ).
A . 0.84
B. 0.54
c. 0.44
D. 0.98
11
803 If Q, B, y, d are the solutions of the equation tan
0+
3 tan 30, no two of which have equal tangents.
4. The value of tan a + tan ß + tan y + tand is
a. 1/3
b. 8/3
c. – 8/3
d. 0
11
804 Prove that ( 1+tan 2 theta tan theta=sec 2 theta ) 11
805 ( boldsymbol{R}=sqrt{boldsymbol{P}^{2} boldsymbol{Q}^{2}-boldsymbol{2} boldsymbol{P} boldsymbol{Q} sin (boldsymbol{theta}-boldsymbol{9} boldsymbol{0})} )
Here ( boldsymbol{P}=mathbf{1 0 1} boldsymbol{g}, boldsymbol{Q}=mathbf{1 0 6} boldsymbol{g}, ) and ( boldsymbol{theta}= )
( mathbf{1 0 9}^{boldsymbol{O}} )
11
806 ( f frac{sqrt{1+sin frac{39 pi}{8}}}{sqrt{1+sin frac{57 pi}{8}}}=tan left(frac{k pi}{16}right) ) then
least positive value of ( k ) is
A . 1
B. 3
( c .5 )
D.
11
807 In the given figure, we have ( A C perp C D, B C perp C D ) and ( D A=D B )
then ( C A=C B ).
A. True
B. False
11
808 The range of ( f(x)=operatorname{cosec}^{2} x+ )
( 25 sec ^{2} x ) is ( (a, infty] ) Find ( a )
11
809 The appropriate value is ( cos 61^{circ} ) is
A . 0.4848
B. 0.4849
c. 0.4948
D. 0.5059
11
810 The general values of ‘ ( theta ) ‘ satisfying the
equation ( sec 4 theta-sec 2 theta=2 ) is
11
811 8. Assume that is a rational multiple of it such that cos A
is a distinct rational. The number of values of cos O is
a. 3
b. 4
c. 5
d. 6
11
812 Convert ( 25^{0} ) into Radian measure. 11
813 69. If cost 0 – sin4 0 = 2, then the
value of 2 cos? 0 – 1 is
(1) 0
(2) 1
11
814 The number of solutions of the equation
( |sin x|=|cos 3 x| operatorname{in}[-2 pi, 2 pi] ) is:
A . 32
B . 28
c. 24
D. 30
11
815 9. Let A = sin x + cos x. Then find the value of sin* x + cos*x
in terms of A.
11
816 e tan x
tan y
= tan 2, x + y + z = n and tanx + tan” y
tan z
5
2
3
+ tan= * then K =
+ tan-z=
then K=
K
11
817 9. The value of tan(a + B) is 11
818 Prove that ( frac{2 tan x}{1+tan ^{2} x}=sin 2 x ) 11
819 Find the value of ( boldsymbol{theta}, ) if
( cot theta=0.2334 )
11
820 17.
The number of solutions of the equation sin(e)* = 5x +5*;
(1990 – 2 Marks
(a) 0
(b) 1
© 2
(d) Infinitely many
11
821 Illustration 3.2 Let A, B, C be the three angles such that
A + B + C = t. If tan A. tan B = 2, then find the value of
cos A cos B
cos C
11
822 If ( a=cos 3 ) and ( b=sin 8 ) then:
( mathbf{A} cdot a>0, b>0 )
B. ( a bb )
D. ( a b>0 )
11
823 ( left(1+tan ^{2} thetaright) cdot sin ^{2} theta= )
( mathbf{A} cdot sin ^{2} theta )
B ( cdot cos ^{2} theta )
( mathbf{c} cdot tan ^{2} theta )
( mathbf{D} cdot cot ^{2} theta )
11
824 If ( A+B+C=180^{circ}, ) then the value of
( (cot B+cot C)(cot C+ )
( cot A)(cot A+cot B) ) will be
( mathbf{A} cdot sec A sec B sec C )
B. ( csc A csc B csc C )
( mathbf{c} cdot tan A tan B tan C )
D.
11
825 ( ln ) a ( Delta A B C, angle A>angle B . ) If ( sin A ) and
( sin B ) satisfy the equation ( 3 sin x- )
( 4 sin ^{3} x-k=0,0<k<1, ) then ( angle C ) is
A ( cdot frac{pi}{3} )
в. ( frac{pi}{2} )
c. ( frac{2 pi}{3} )
D. ( frac{5 pi}{6} )
11
826 6. Ifsin-x – 2 sinx-1=0 has exactly four different solutions
in xe [0, nn), then value/values of n is/are (n e M)
b. 3
c. 4
d. 6
a. 5
11
827 98. If o, B, y are acute angles and cos 0 = sin B/sin a, cos
= sin / sin a and cos (0 – 0) = sin ß sin y, then the value
of tan’a – tanºß – tany is equal to
a. – 1
b. 0
2 . 1
11
828 Find the value of ( 5 sin 30^{0}+3 tan 45^{0} ) 11
829 If ( tan alpha ) and ( tan beta ) are the roots of the
equation ( boldsymbol{x}^{2}+boldsymbol{p} boldsymbol{x}+boldsymbol{q}=boldsymbol{0}(boldsymbol{p} neq boldsymbol{0}), ) then
This question has multiple correct options
A ( cdot sin ^{2}(alpha+beta)+p sin (alpha+beta) cos (alpha+beta)+q cos ^{2}(alpha+beta)= )
( q )
B. ( tan (alpha+beta)=frac{p}{q-1} )
( mathbf{c} cdot cos (alpha+beta)=1-q )
( mathbf{D} cdot sin (alpha+beta)=-p )
11
830 Express in Degrees:
( (a)left(frac{2 pi}{15}right)^{c} )
( (b)(-2)^{c} )
A ( cdot(a) 14^{circ} )
( (b) 244^{circ} 32^{prime} 44^{prime prime} )
B . ( (a) 74^{circ} )
( (b)-114^{circ} 32^{prime} 44^{prime prime} )
c. ( (a) 14^{circ} )
( (b)-120^{circ} 32^{prime} 44^{prime prime} )
D cdot ( (a) 24^{circ} )
( (b)-114^{circ} 32^{prime} 44^{prime prime} )
11
831 The value of ( sin frac{theta}{2} cdot sin frac{7 theta}{2}+ )
( sin frac{3 theta}{2} cdot sin frac{11 theta}{2}-sin 2 theta cdot sin 5 theta ) is equal
to
( mathbf{A} cdot mathbf{0} )
B.
( mathbf{c} cdot sin 6 theta-cos 5 theta )
( D cdot sin 6 theta+sin 7 theta )
11
832 20. A right angle is divided into three positive parts a, B and
y. Prove that for all possible divisions tana + tan ß +
tan y> 1 + tan a tan ß tan y.
11
833 Illustration 4.14 If the equation a sinx + cos2x = 2a – 7
possesses a solution, then find the values of a.in
11
834 If ( boldsymbol{m} tan left(boldsymbol{theta}-mathbf{3 0}^{circ}right)=boldsymbol{n} tan left(boldsymbol{theta}+mathbf{1 2 0}^{circ}right) )
then ( frac{m-n}{m+n} ) is equal to
A ( .2 cos 2 theta )
B. ( 2 sin ^{2} theta )
c. ( 1 /(8 cos 2 theta) )
D. ( 1 /(8 sin 2 theta) )
11
835 49. If tan²0=2 tanềo + 1, then cos 20+ sin-o equals
a. -1
b. 0
c. 1
d. none of these
11
836 then –
2 sin e
15. If x=
1+cos + sino’
a. 1 + x
c. x
1-cos + sin e
– is equal to
1+ sin o
b. 1-X
d. 1/8
11
837 Illustration 4.16
Solve 2 cos²0+ 3 sin 0 = 0.
11
838 f ( tan theta+sin theta=m ) and ( tan theta-sin theta= )
( n, ) then prove that ( m^{2}-n^{2}=4 sqrt{m n} )
11
839 19. IfX=sin ( 0 +7.) + sin( 0 – 6 )+ sin( 0 + 32)
Y=cos 0 +) = cos (6-7) + cos( 0 + ?
Y
3 X
then prove that —
Y
= 2 tan 20.
X
11
840 Evaluate each of the following in the
simplest form:
( cos 60^{circ} cos 30^{circ}-sin 60^{circ} sin 30^{circ} )
11
841 Illustration 3.59 Prove that
Vsinx+4 cos²x – cos*x+4 sin²x = cos 2x.
11
842 If u = Vacose + b2 sine + Vasino + b2 cos?
then the difference between the maximum and minimum
values of – is given by
12004)
(a) (a – b)2
(b) 2a + b2
(c) (a+b)
(d) 2(a? +62)
11
843 Value of ( tan 15^{circ} ) is
This question has multiple correct options
A ( cdot frac{sqrt{3}-1}{sqrt{3}+1} )
B. ( 2-sqrt{3} )
( c cdot 2+sqrt{3} )
D. ( sqrt{3}-1 )
11
844 The value of ( frac{6 pi^{c}}{5} ) in sexagesimal
measure is
A . ( 144^{circ} )
B . ( 216^{circ} )
( c cdot 240^{9} )
D. ( 120^{9} )
11
845 If ( A+B+C=pi ) then which of the
following are true?
i. ( tan 3 A+tan 3 B+tan 3 C= )
( tan 3 A tan 3 B tan 3 C )
ii. ( cot frac{A}{2}+cot frac{B}{2}+cot frac{C}{2}= )
( cot frac{A}{2} cot frac{B}{2} cot frac{C}{2} )
A. only 1st statement is true
B. only 2nd statement is true
c. both statements are true
D. both statements are false
11
846 If sin a = A sin(a+ß), A #0, then
1. The value of tan a is
Asin B
Asin ß
a.
b. –
1- Acos B
1+ Acos ß
A cos
B
d .
Asin ß
1 – Asin
B
1 + Acos ß
11
847 Solve:
( left(sqrt{3}+tan 1^{0}right) )
11
848 Solve ( tan theta+tan 2 theta+ )
( sqrt{3} tan theta tan 2 theta=sqrt{3} )
A. ( theta=frac{n pi}{3}+frac{pi}{3}, n in Z )
в. ( theta=frac{n pi}{3}+frac{pi}{6}, n epsilon Z )
c. ( theta=frac{n pi}{3}+frac{pi}{12}, n in Z )
D. ( theta=frac{n pi}{3}+frac{pi}{9}, n in Z )
11
849 Illustration 3.98 If x, y e R and x2 + y2 + xy = 1, then find
the minimum value of xy + xy + 4.
11
850 The value of ( frac{4}{tan ^{2} 60^{0}}+frac{1}{cos ^{2} 30^{0}}- )
( sin ^{2} 90^{0} ) is equal to
11
851 Illustration 3.67 Find the value of cos 12° + cos 84° +
cos 156° + cos 132º.
11
852 5.
The general solution of the trigonometric equation sin x+cos
x=1 is given by :
(1981 – 2 Marks)
(a) x = 2nt ; n=0, #1, #2…
(b) x = 2n1 + n/2;n=0, +1, 2…
(C) x=nn+(-1)”
(d) none of these n=0, +1, +2…
11
853 Illustration 3.15
Prove that –
cos 10° + sin 10°
-= tan 55º.
cos 10º – sin 10°
11
854 Assertion ( f(x)=frac{2}{pi} x sin x+x^{3}, ) where ( x in )
( left[0, frac{pi}{2}right] )
Statement-1: ( f(x)=frac{pi}{2} ) has exactly one solution in ( boldsymbol{x} inleft[mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right] )
and
Reason
Statement-2: ( boldsymbol{f}(boldsymbol{x}) geq mathbf{0} ) for all ( boldsymbol{x} ) in
( left[0, frac{pi}{2}right] )
A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
c. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.
11
855 Illustration 3.28 If sin A = sin B and cos A = cos B, then
A B a
prove that sin = 0.
11
856 Which one of the following is not correct
( ? )
A ( cdot frac{(sin A+sin B)}{(sin A-sin B)}=frac{tan frac{1+beta}{2}}{tan frac{1-B}{2}} )
( mathbf{B} cdot sin ^{2} A-cos ^{2} B=sin (A+B) sin (A-B) )
C ( cdot cos A-cos B=2 cos frac{A+B}{2} cos frac{B-A}{2} )
( mathbf{D} cdot cos ^{2} A-cos ^{2} B=sin (A+B) sin (B-A) )
11
857 Illustration 4.26
Solve sec 40 – sec 20= 2.
11
858 Illustration 4.8 If sin A = sin B and cos A = cos B, then find
the value of A in terms of B.
11
859 For any ( theta ), state the value of:
( sin ^{2} theta+cos ^{2} theta )
11
860 Illustration 4.41 Solve logtan (2 + 4 cos²x) = 2. 11
861 If ( A, B ) are acute angles, ( sin A= ) ( frac{4}{5}, tan B=frac{5}{12} ) then ( sin (A+B)= )
A ( cdot frac{36}{65} )
в. ( frac{65}{56} )
c. ( frac{65}{63} )
D. ( frac{63}{65} )
11
862 Suppose ( x ) and ( y ) are real numbers such that ( tan x+tan y=42 ) and ( cot x+ )
( cot y=49, ) then the single digit prime
number by which the value of
( tan (x+y) ) is not divisible is
11
863 If ( frac{x}{y}=frac{cos A}{cos B} operatorname{then} frac{x tan A+y tan B}{x+y}= )
A. ( cot frac{A+B}{2} )
B. ( cot frac{A-B}{2} )
c. ( tan frac{A-B}{2} )
D. ( tan frac{A+B}{2} )
11
864 14. If 2tan´x – 5sec x = 1 is satisfied by exactly seven
distinct values of xe |,ne N, then the
2
greatest value of n is
11
865 Simplify:
( frac{1-cos x}{1+cos x} )
11
866 64. If A is an acute angle and cot A+
cosec A = 3, then the value of
sin A is
(1) 1
(2)
(4) O
11
867 The number of solutions of the equation ( |cot x|=cot x+frac{1}{sin x}(0 leq x leq 2 pi) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
( D )
11
868 Prove that:
[
cos 20^{circ}+cos 100^{circ}+cos 140^{circ}=0
]
11
869 53. One of the general solutions of 13 cos 0 – 3 sin 0 = 4
sin 20 cos 30 is
a. mn + /18, me Z b. mtt/2 + Tt/6, V me Z
c. m m/3 + 7/18, me Z d. none of these
11
870 Prove that: ( frac{sin A-2 sin ^{3} A}{2 cos ^{3} A-cos A}=tan A ) 11
871 Illustration 3.81 If A
tan A
tan B
tan B.tan C tan A. tan C
– 2cotA – 2cot B – 2cotC
+ B + C = 1, prove that
tan C
=tan A + tan B + tan C
tan A. tan B
11
872 49. The set of values of x satisfying the equation sin 3a = 4
sin a sin(x + a) sin(x – a) is
a. nn + Tt/4, v nez b. nnt + 7/3, Vnez
c. na = 7/9, Vnez d. ntt + Tt/12, Vnez
11
873 6. The general values of 0 satisfying the equation 2 sin’e-3
sin 0-2 = 0 is (n e Z)
a. nt+ (-1)” 7/6 b. TTC+(- 1)” T/2
c. nt+ (-1)” 570/6 d. nnt + (-1)” 770/6
11
874 20. 272 – 1 is equal to
a. sin a
c. sin
b. cos a
d. cos e
11
875 Prove that:
( frac{sin A-2 sin ^{3} A}{2 cos ^{3} A-cos A}=tan A )
11
876 Illustration 4.2
Solve
tan 3x – tan 2x
1 + tan 3x tan 2x
= 1.
11
877 If ( sin alpha+operatorname{cosec} alpha=2, ) find the value of
( sin ^{n} alpha+operatorname{cosec}^{n} alpha, n epsilon Z )
11
878 Prove that ( frac{sin 60^{circ}+sin 30^{circ}}{sin 60^{circ}-sin 30^{circ}}= )
( frac{tan 60^{circ}+tan 45^{circ}}{tan 60^{circ}-tan 45^{circ}} )
11
879 Find the acute angles ( A ) and ( B ) satisfying ( sec A cot B-sec A-2 cot B+2=0 ) 11
880 Illustration 3.1 Prove that
sin (B-C) sin(C – A)
cos B cos C cos Ccos A
sin(A-B)
cos A cos B = 0.
11
881 ( f-1+cos 56^{circ}+cos 58^{circ}+cos 66^{circ}= )
( k sin 28^{circ} sin 29^{circ} sin 33^{circ}, ) then the value
of k is
A . 1
B. 3
( c cdot 4 )
D. 5
11
882 Illustration 3.80 If A + B + C = Tt, prove that cot A + cot B
+ cot C – cosec A . cosec B. cosec C = cot A . cot B. cot C.
11
883 If ( tan (A+B)=1, ) and ( cos (A-B)= )
( frac{sqrt{3}}{2}, 0^{o}<A+BB ; ) find
( A ) and ( B )
11
884 Illustration 3.75 If A + B + C = 180°, prove that cos? A +
cos B + cos-C=1 – 2 cos A cos B cos C.
11
885 cot 25° + cot 55°
Illustration 3.20 Find the value of –
tan 25° + tan 550 +
cot 55° + cot100°
tan 55° + tan 100°
cot100° + cot 25°
tan 100° + tan 25°
11
886 ( sin 85^{circ}-sin 35^{circ}-cos 65^{circ}= )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
11
887 The value of ( cos ^{2} 30^{0}-cos ^{2} 60^{0}- )
( cos 60^{0} ) is
( mathbf{A} cdot mathbf{0} )
в. ( frac{1}{2} )
( c cdot frac{3}{4} )
D.
11
888 The number of solutions of ( 3 sec theta-5= )
( 4 tan theta operatorname{in}[0,4 pi] ) be ( k . ) Find ( k ? )
11
889 12-sin a – cos a in
is equal to
sin a – cos a
α
π
a. sec
b. cos
Bloo
o
2
a
c. tan
Bloo
d. cot
11
890 Find the general solution of ( sec theta+1= ) ( (2+sqrt{3}) tan theta ) 11
891 The set of values of a for which the
equation ( sin x(sin x+cos x)=a ) has
real solutions is
A ( cdot[1-sqrt{2}, 1+sqrt{2}] )
B. ( [2-sqrt{3}, 2+sqrt{3} )
D. ( left[frac{1-sqrt{2}}{2}, frac{1+sqrt{2}}{2}right] )
11
892 The value of ( cos ^{4}left(frac{pi}{4}right)-cos ^{4}left(frac{pi}{6}right)+ )
( sin ^{4}left(frac{pi}{6}right)+sin ^{4}left(frac{pi}{3}right) ) is
A ( cdot frac{1}{16} )
в. ( frac{1}{8} )
c. ( frac{5}{16} )
D. ( frac{3}{16} )
11
893 Find the general solution:
( sin x+sin 3 x+sin 5 x=0 )
11
894 5. For all in [0, r/2] show that cos(sin O) sin(cos 6).
CITLIFE 1981
11
895 The value of ‘ ( b^{prime} ) such that the equation ( frac{b cos x}{2 cos 2 x-1}=frac{b+sin x}{left(cos ^{2} x-3 sin ^{2} xright) tan x} )
possess solution, then prove that belongs to the set ( left(-infty, frac{1}{2}right) )
11
896 If ( frac{sin ^{3} theta-cos ^{3} theta}{sin theta-cos theta}-frac{cos theta}{sqrt{1+cot ^{2} theta}}- )
( 2 tan theta cot theta=-1, theta in[0,2 pi], ) then
A ( cdot theta inleft(0, frac{pi}{2}right)-left{frac{pi}{4}right} )
в. ( _{theta} inleft(frac{pi}{2}, piright)-left{frac{3 pi}{4}right} )
c. ( _{theta inleft(pi, frac{3 pi}{2}right)-left{frac{5 pi}{4}right}} )
D. ( theta in(0, pi)-left{frac{pi}{4}, frac{pi}{2}right} )
11
897 if ( boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon}[mathbf{0}, boldsymbol{2} boldsymbol{pi}] ) and ( sin boldsymbol{x}+sin boldsymbol{y}=boldsymbol{2} ) then
the value of ( x+y ) is
A . ( pi )
в.
( c .3 pi )
D. None of these
11
898 ( left{mathbf{x} in mathbb{R}: cos 2 x+2 cos ^{2} x-2=0right}= )
A ( cdotleft{2 n pi+frac{pi}{3}, n in Zright} )
в ( cdotleft{n pi pm frac{pi}{6}, n in Zright} )
с ( cdotleft{n pi+frac{pi}{3}, n in Zright} )
D ( cdotleft{2 n pi-frac{pi}{3}, n in Zright} )
11
899 Illustration 3.6 If 3tan 8 tan o = 1, then prove that
2 cos(0+ 0) = cos(0-0).
11
900 Statement
(I): If
( sin alpha=frac{12}{13},left(0<alpha<frac{pi}{2}right) ) and
( cos beta=-frac{3}{5},left(pi<beta<frac{3 pi}{2}right) ) then
( sin (alpha+beta)=frac{56}{65} )
Statement (II): If
( theta ) and ( phi ) are angles in the first quardrant such that ( tan theta=frac{1}{7} ) and ( sin phi=frac{1}{sqrt{10}} operatorname{then} theta+2 phi=45^{circ} )
Which of the above statements is
correct?
A. only
B. only ॥
c. Both I&॥
D. Neither I nor I
11
901 If ( tan alpha=K cot beta, ) then ( frac{cos (alpha-beta)}{cos (alpha-beta)} )
equals
A ( cdot frac{1+K}{1-K} )
в. ( frac{1-K}{1+K} )
c. ( frac{K+1}{K-1} )
D. ( frac{K-1}{K+1} )
11
902 Illustration 3.5 Show that cos 0 + cos²(a + ) – 2 cos a
cos e cos(a + 0) is independent of e.
11
903 Find the value of ( sin 765^{circ} ) 11
904 if ( alpha ) and ( beta ) are angles in the first quadrant, ( tan alpha=frac{1}{7}, sin beta=frac{1}{sqrt{10}}, ) then
using the formula ( sin (A+B)= )
( sin A cos B+cos A sin B, ) one can find
the value of ( (boldsymbol{alpha}+mathbf{2} boldsymbol{beta}) ) to be
( A cdot 0^{circ} )
B. 45
( c cdot 60 )
( D cdot 90^{circ} )
11
905
23. The expression cos 30 + sin 30 + (2 sin 20 – 3) (sin
cos ) is positive for all in
a. (ann – 31 , 2n + 7), nez
b. (2nt – , 2nn + ) nez
4
11
906 if ( sec theta+tan theta=x, ) then what is the
value of ( sec theta ? )
11
907 10. Show that the value of , wherever defined, never
tan 3x
lies between – and 3.
(IIT-JEE 1992)
11
908 ( f cos ^{2} 2 x-cos ^{2} 6 x=sin m x sin 8 x )
Find ( boldsymbol{m} )
11
909 1. Prove that
sin x – cos x +1
sin x + cos x -1
= secx + tan x.
11
910 Illustration 2.20
Express 1.2 rad in degree measure.
11
911 ( cos 20^{circ} cos 100^{circ}+cos 100^{circ} cos 140^{circ}- )
( cos 140^{circ} cdot cos 200^{circ} ) is equal to
( A cdot frac{3}{4} )
B. ( frac{1}{4} )
( c cdot-frac{1}{4} )
D. ( -frac{3}{4} )
11
912 If ( frac{tan 2 theta+tan theta}{1-tan theta tan 2 theta}=0, ) then the
general value of ( boldsymbol{theta} ) is
A ( , n pi ; in I )
( I )
B. ( frac{n pi}{3} ; in I )
c. ( frac{n pi}{4} ; in I )
D. ( frac{n pi}{6} ; in I )
11
913 ( sin 360^{circ}=? ) 11
914 6. Let e, OE 10, 27t] be such
that 2 cos 0 (1 – sin
) =
sin e tan
+cot
cos 0 – 1, tan(21 – 0) > 0 and
-1<sin
. Then q cannot satisfy
(IIT-JEE 2012)
11
915 ( frac{sin (alpha+beta)}{sin (alpha-beta)}=frac{a+b}{a-b} ) then prove that
( boldsymbol{a} tan boldsymbol{beta}=boldsymbol{b} tan boldsymbol{alpha} )
11
916 Given ( tan (pi cos theta)=cot (pi sin theta) )
then the value of ( cos left(theta-frac{1}{4} piright) ) will be
( ^{A} cdot frac{1}{2 sqrt{2}} )
B. ( frac{1}{sqrt{2}} )
c. ( frac{1}{3 sqrt{2}} )
D. ( frac{1}{4 sqrt{2}} )
11
917 Illustration 4.58
Find the number of solutions of sinx= –
11
918 If ( tan A=frac{1}{2}, tan B=frac{1}{3} ) then
( tan (A+B)= )
A .
в.
c. -1
D.
11
919 1.
m
and tanſ =
find the possible values
If tana=-
m+1
of(a+b).
2m +1
(1978)
11
920 21. Which of the following set of values of x satisfies the
equation 2(2 sin x-3 sin x+1) + 2(2-2 sin? x +3 sin x) = 9?
TT
a. X=n7+
,ne I
b. x=nnt
,ne I
c. x=nt, ne
I
d
. x= 2n1+
,ne I
11
921 60. If 4 sin 20+ sin²0 = 4, then what
will be the value of tan (90° + 0)
from the following ?
(1) O
11
922 13. Show that the equation esinx-e-sin x – 4 = 0 has no real
solution.
(1982 – 2 Marks)
11
923 1. Number of values of p for which equation sin’x + 1 +
p3 – 3 p sinx = 0 (p > 0) has a root is
11
924 Find the value of
( cos 1^{0} cos 2^{0} cos 3^{0} ldots . . cos 89^{0} ldots . . cos 179 )
11
925 Solve the following equations.
( cos 9 x-2 cos 6 x=2 )
11
926 If ( alpha ) is only real root of the equation
( boldsymbol{x}^{3}-(cos 1) boldsymbol{x}^{2}+(sin 1) boldsymbol{x}+1=mathbf{0}, ) then
( left(tan ^{-1} alpha+tan ^{-1} frac{1}{alpha}right) ) cannot be equal
to
his question has multiple correct options
( mathbf{A} cdot mathbf{0} )
в. ( frac{pi}{2} )
( c cdot-frac{pi}{2} )
D.
11
927 Prove that ( cos ^{2} 45^{circ}-sin ^{2} 15^{circ}=frac{sqrt{3}}{4} ) 11
928 The number of solutions of the equation ( 1+sin ^{4} x=cos ^{2} 3 x, x inleft[-frac{5 pi}{2}, frac{5 pi}{2}right] )
is?
( mathbf{A} cdot mathbf{5} )
B. 4
( c cdot 7 )
( D )
11
929 ( sin left(frac{pi}{2}-xright)=cos x ) 11
930 22. In a right angled triangle, acute angles A and B satisfy
tan A + tan B + tan-A + tan-B + tan A + tanB = 70.
Find the angle A and B in radians.
11
931 In an Isosceles triangle ( A B C, tan ^{2} B- )
( sec ^{2} B+2 )
11
932 If ( mathrm{B} ) be the exterior angle of a regular
polygon of ( n ) sides and ( A ) is any constant, then prove that ( sin A+sin (A+B)+sin (A+2 B)+ )
( . . n ) terms ( =0 )
11
933 The number of solutions of the equation ( frac{sec x}{1-cos x}=frac{1}{1-cos x} ) in ( [0,2 pi] ) is
equal to
( A cdot 3 )
B. 2
( c )
( D )
11
934 Illustration 4.13 Find the number of solutions of the
equation esinx – e-sinx – 4 =0.
11
935 Prove that ( : frac{1}{sin 10^{circ}}-frac{sqrt{3}}{cos 10^{circ}}=4 ) 11
936 In a circle of diameter ( 40 mathrm{cm} ), the length
of a chord is ( 20 mathrm{cm} . ) Find the length of minor arc of the chord.
11
937 c
10. Let cos(a+b) = andsin (a–B) = 13 where
osa,Bs.Then tan 2a =
[2010]
11
938 then
17. If y = (1 + tan A) (1 – tan B), where A – B
(y + 1)p+l is equal to
zista.9
b. 4
c. 27
d. 81
11
939 The value of expression ( frac{(2 tan 4+3 cot 4)(2 cot 4+3 tan 4)}{24 cot ^{2} 8+25} ) is
( A )
B. 2
( c cdot 3 )
( D )
11
940 If ( sin theta=-frac{4}{5}, pi<theta<frac{3 pi}{2}, ) then find
i) ( sin 2 theta )
ii) ( cos 2 theta )
iii) ( tan 2 theta )
11
941 What is the ( sin (boldsymbol{alpha}+boldsymbol{beta})- )
( 2 sin alpha cos beta+sin (alpha-beta) ) equal to?
( mathbf{A} cdot mathbf{0} )
B. ( 2 sin alpha )
( c cdot 2 sin beta )
( D cdot sin alpha+sin beta )
11
942 Illustration 4.18
Solve sin
cos
– cos O sin 0= 1/4.
11
943 ( sec ^{2} 50^{circ}-cot ^{2} 40^{circ}-sin ^{2} 45^{circ}=? )
A ( cdot 1 / 2 )
B. 5
( c cdot 1 )
D.
11
944 Find the radian measures
corresponding to the following degree
measures:
( 25^{circ} )
( -47^{circ} 30 )
( 240^{circ} )
( 520^{circ} )
11
945 39
6. Suppose a, and B, are the roots of the
12
equation x2 – 2x sec a + 1 = 0 and a, and B, are the roots of
the equation x2 + 2x tan 0-1=0.Ifa, >, and > B2, then
aj + B2 equals
(JEE Adv. 2016)
(a) 2 (sec -tan )
(b) 2 sec
(c) -2 tano
(d) 0
11
946 19. The number of solution of sinºx – cos²x sin x + 2 sin’x +
sin x = 0 in 0 SX S 31 is
a. 3
b.4
d. 6 )
c. 5
11
947 Find the range if ( [2 sin x]+[cos x]= )
( -3, ) then the range of the function
( f(x)=sin x+sqrt{3} cos x ) in ( [0,2 pi] )
(where [.] denotes the greatest integer function)
A ( cdot(2,-1) )
B ( cdotleft(-1,-frac{1}{2}right) )
( mathbf{c} cdot(-2,-1) )
D. None of these
11
948 35. Let A = sin80+ cos14e; then for all real e
a. A 21
b. 0<ASI
c. -<AS-
d. none of these
11
949 Prove that ( frac{sin x-sin 3 x}{sin ^{2} x-cos ^{2} x}=2 sin x ) 11
950 Find the principal and general solutions of the following equation:
( sec x=2 )
A ‘ principal solution ( =frac{pi}{6}, frac{5 pi}{6} ) and General solution ( = )
( n pi pm frac{pi}{6}, n in Z )
B. Principal solution ( =frac{pi}{3}, frac{5 pi}{3} ) and General solution ( = )
( n pi pm frac{pi}{3}, n in Z )
C ‘ principal solution ( =frac{pi}{6}, frac{5 pi}{6} ) and General solution ( = )
( 2 n pi pm frac{pi}{6}, n in Z )
D. principal solution ( =frac{pi}{3}, frac{5 pi}{3} ) and General solution ( = )
( 2 n pi pm frac{pi}{3}, n in Z )
11
951 A lamp post stands on a horizontal plane. From a point situated at a distance ( 150 mathrm{m} ) from its foot, the angle
of elevation of the top is ( 30^{circ} . ) What is the
height of the lamp post?
A . ( 50 m )
в. ( 50 sqrt{3} m )
c. ( frac{50}{sqrt{3}} m )
D. ( 100 m )
11
952 Prove that ( (operatorname{cosec} theta-cot theta)^{2}=frac{1-cos theta}{1+cos theta} ) 11
953 73. cos’x sin 2x = a, sin(rx) Vxe R, then
x=0
a. n= 5, a, = 1/2 b. n=5, a, = 1/4
c. n=5, a, = 1/8 d. n=5, a, = 1/4
11
954 68. Given that (1+11+x) tan y =1+v1- x. Then sin 4y is
equal to
a. 4x
b. 2x
c. X
d. none of these
11
955 If ( sin theta=frac{1}{2}, ) show that ( (3 cos theta- )
( left.4 cos ^{3} thetaright)=0 )
11
956 Number of value of ( boldsymbol{x} in[mathbf{0}, mathbf{4} boldsymbol{pi}] ) and
satisfying ( sqrt{2} sec x+tan x=1 ) is?
11
957 Illustration 2.4
Convert 45° to radians.
11
958 Prove that ( y=frac{4 sin theta}{(2+cos theta)}-theta ) is an
ncreasing function of ( boldsymbol{theta} ) in ( left[mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right] )
11
959 58. The number of values of 0 satisfying 4 cos 0 + 3 sin 0 =
5 as well as 3 cos + 4 sin = 5 is
a. one
b. two
c. zero
d. none of these
11
960 5. If sin 0, + sin 02 + sin 02 = 3, then cos 6, + cos Oz + cos 03
is equal to
a. 3
b. 2
c. 1
do o
11
961 R
7. If A, B, C are angles of a triangle, then 2sin
sin – sin A cot – cos A is
100
= (0)
a. independent of A, B, C 100
b. function of A, B
c. function of C
d. none of these
11
962 Illustration 2.53 Solve tan x > cotx, where x € [0,21]. 11
963 Illustration 3.68 Prove that
cos 36° cos 72° cos 108° cos 144° = 1/16.
11
964 An angle which is equal to ( 360^{circ} ) is called ( _{text {一一一一一一一 }} ) angle.
A. Right
B. Complete
c. Acute
D. obtuse
11
965 1. In a AABC, if tan A/2, tan B/2, tan C/2 are in A.P., then
show that cos A, cos B, cos C are in A.P.
11
966 f ( cos (alpha+beta) sin (gamma+theta)=cos (alpha- )
( beta) sin (gamma-theta) . ) show that ( tan theta= )
( tan alpha tan beta tan gamma )
11
967 The value of ( frac{tan 45^{circ}}{sin 30^{circ}+cos 30^{circ}} ) is:
This question has multiple correct options
A. ( frac{2}{sqrt{3}+1} )
B.
( frac{1+sqrt{3}}{2} )
D. ( sqrt{3}-1 )
11
968 Eliminate ( x ) from equations
( sin (a+x)=2 b ) and ( sin (a-x)=2 c )
A ( cdot frac{(b+c)^{2}}{sin ^{2} a}-frac{(b-c)^{2}}{cos ^{2} a}=1 )
B. ( frac{(b+c)^{2}}{sin ^{2} a}+frac{(b+c)^{2}}{cos ^{2} a}=1 )
( ^{C} cdot frac{(b+c)^{2}}{sin ^{2} a}+frac{(b-c)^{2}}{cos ^{2} a}=1 )
D. ( frac{(b+c)^{2}}{sin ^{2} a}+frac{(b-c)^{2}}{cos ^{2} a}=-1 )
11
969 If ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi}, ) show that ( sin left(frac{boldsymbol{A}}{mathbf{2}}right)+ )
( sin left(frac{B}{2}right)+sin left(frac{C}{2}right)=1+ )
( 4 sin frac{pi-A}{4} sin frac{pi-B}{4} sin frac{pi-C}{4} )
11
970 65. The sum of all the solutions in [0, 41] of the equation
tanx + cotx + 1 = cos(x+is
a. 37
c. 77/2
b. /2
d. 411
11
971 The value of ( 60^{g} ) in circular measure is
A ( cdot frac{pi^{c}}{10} )
в. ( frac{3 pi^{c}}{10} )
c. ( frac{2 pi^{c}}{5} )
D. ( frac{pi^{c}}{2} )
11
972 Illustration 4.64
Solve sin 0+ V3 cos 021, -< OST.
11
973 Which of the following statements are
possible; ( a, b, m ) and ( n ) being non-zero
real numbers?
A ( cdot 4 sin ^{2} theta=5 )
B ( cdotleft(a^{2}+b^{2}right) cos theta=2 a )
C. ( left(m^{2}+n^{2}right) csc theta=m^{2}-n^{2} )
D. ( sin theta=2.375 )
11
974 Assertion
If ( tan A+tan B+tan C=3 sqrt{3}, ) then
triangle is equilateral
Reason
( ln Delta A B C, tan A+tan B+tan C= )
( tan A tan B tan C )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion false but Reason is true
11
975 13. The value of cosec10° +cosec50° – cosec 70° is 11
976 7. Suppose A and B are two angles such that A, B € (0, 1),
and satisfy sin A + sin B = 1 and cos A +cos B = 0. Then
the value of 12 cos 2A + 4 cos 2B is
11
977 27. The number of real roots of the equation cosec 0 +
sec 0 – 115 = 0 lying in [0, 1] is
a. 6
b. 8
d. 0
c. 4
11
978 Prove that
[
frac{sin (theta+phi)-2 sin theta+sin (theta-phi)}{cos (theta+phi)-2 cos theta+cos (theta-phi)}=
]
( tan theta )
11
979 Prove ( left(frac{1+tan ^{2} A}{1+cot ^{2} A}right)=left(frac{1-tan A}{1-cot A}right)^{2} ) 11
980 Illustration 3.9 In a triangle ABC, if sin A sin (B-C)=sin C
sin (A – B), then prove that cot A, cot B, cot C are in A.P..
11
981 nt
| 70. If 12 sin 0 – cosec 6 2 1 and Ot **,ne Z, then
COS
a. cos 20 2 1/2
c. cos 20 1/4
d. cos 20 < 1/4
11
982 If ( cos alpha+cos beta=0=sin alpha+sin beta )
then ( cos 2 alpha+cos 2 beta ) is equal to
A ( .-2 sin (alpha+beta) )
B. ( -2 cos (alpha+beta) )
( mathbf{c} cdot 2 sin (alpha+beta) )
D. ( 2 cos (alpha+beta) )
11
983 If ( 0 leq x leq pi ) and ( 81^{sin ^{2} x}+81^{cos ^{2} x}=30 )
then ( x ) is equal to:
A ( cdot frac{pi}{6} )
в. ( frac{pi}{2} )
c.
D. ( frac{3 pi}{4} )
11
984 15. If cos x – sin a cot B sin x = cos a, then the value of
tan (x/2) is
a. –tan (a/2) cot (B/2) b. tan (a/2) tan (B/2)
att bo arc. – cot (0/2) tan (B/2) d. cot (0/2) cot (B/2)
11
985 If ( sec theta=sqrt{2} ) and ( theta ) lies in first
quadrant. Find value of ( frac{1+tan theta+operatorname{cosec} theta}{1+cot theta-operatorname{cosec} theta} )
11
986 sina, cos aan
cos a and
0.
The sides of a triangle are
V1+sin a cosa for some 0<a Then the greatest
angle of the triangle is
[2004]
(a) 150° (6) 90° (c) 120° (d) 60°
hot the
11
987 65. sec 50°. sin 40° +
cos 40°. cosec 50° = ?
(1) 2
(2) O
(3) 1 (4) 15
11
988 Which of the following statements are
correct?
This question has multiple correct options
( mathbf{A} cdot sin 1>sin 1^{circ} )
B. ( tan 2tan 2 )
D. ( tan 2<tan 1<0 )
11
989 ( (operatorname{cosec} A-sin A)(sec A-cos A)= )
( frac{1}{tan A+cot A} )
[Hint: Simplify LHS and RHS separately]
11
990 5. In triangle ABC, if sin A cos B = 2 and 3 tan A = tan B,
then cotA is equal to
b. 3
c. 4
a. 2
d. 5
:
11
991 5. sin 0 + 73 cos 0 = 6x – x? – 11,0 5 Os 41, x € R, holds
for
a. no values of x and O
b. one value of x and two values of 0
c. two values of x and two values of O
d. two point of values of (x, 0)
11
992 Illustration 3.56 If sin a + sin ß= a and cos a + cos ß= b,
prove that tan a – B-+ 4-a? – b?
az + 12
11
993 41. The roots of the equation 4×2 – 2 15 x + 1 = 0 are
a. sin 36°, sin 18° b. sin 18°, cos 36°
c. sin 36º, cos 18° odd. cos 18°, cos 36°
11
994 Illustration 4.12 If xe (0, 21) and ye (0, 2), then find the
number of distinct ordered pairs (x, y) satisfying the equation
9 cos²x + sec’y – 6cos x – 4 sec y + 5 = 0.
11
995 Solve for ( x: sin x+sin 2 x+sin 3 x=3 )
where ( x in(0, pi) )
11
996 The value of ( sqrt{2}left(cos 15^{circ}-sin 15^{circ}right) ) is
equal to:
A ( cdot sqrt{3} )
B. ( sqrt{2} )
c. 1
D. 2
E. ( 2 sqrt{3} )
11
997 Solve:
( x sin 45^{circ} cdot cos 45^{circ} cdot tan 60^{circ}=tan 45^{circ} )
( cos 60^{circ} )
11
998 8.
Which of the following number(s) is/are rational?
(1998 – 2 Marks)
(a) sin 15°
(b) cos 15°
c) sin 15° cos 150
(d) sin 15° cos 75°
11
999 Prove ( sin frac{8 pi}{3} cos frac{23 pi}{6}+ )
( cos frac{13 pi}{3} sin frac{35 pi}{6}=frac{1}{2} )
11
1000 The solution of ( (s e c boldsymbol{theta}+1)=(2+ ) ( sqrt{3}) tan theta(0<theta<2 pi) ) are
( mathbf{A} cdot pi / 6, pi )
в. ( pi / 3, pi / 4 )
c. ( pi / 6,2 pi / 3 )
D. none of these
11
1001 In triangle ( A B C, ) right-angled at ( B ), if
( tan A=frac{1}{sqrt{3}}, ) find the value of:
(i) ( sin A cos C+cos A sin C )
(ii) ( cos A cos C-sin A sin C )
11
1002 If ( tan left(45^{circ}+thetaright)=sqrt{3} ) and ( 0<theta<20^{circ} )
then the value of ( theta ) is
A ( .10^{circ} )
B . ( 15^{circ} )
( c cdot 20 )
D. ( 5^{circ} )
11
1003 15. If – sin 0, cos 0, tan 0 are in G.P., then e is equal to
(ne 2)
a. 2n
+
b. 2nnt I
c. NT +(-1)”
d. nt +
11
1004 If ( 3 sin theta+4 cos theta=5, ) then the value of
( 4 sin theta-3 cos theta ) is
( A cdot 0 )
B.
( c .5 )
D. none of these
11
1005 Illustration 3.60
= tan 90
Prove that (4 cos29º – 3) (4 cos 27° – 3)
11
1006 63. If =kk #1) then which of the following is not
tan A
true?
A cos A k-1
sin 3 A 2kl 2
a.
cos3A 2
s in A k-1 mall
cot 3 A 1
==
d. none of these
cot A k
11
1007 2 sin 20
29. Given that, Ano) = –
cos 20 – cos 4ne, and fO) + f(20) +
sin 20
(30) + … +f(no) = ?
sin sine, then the value of u-2 is
11
1008 In a ( triangle A B C, angle A ) is greater than ( angle B ). If
the measures of ( angle A ) and ( angle B ) satisfy the
equation ( 2 tan x-kleft(1+tan ^{2} xright)=0 )
where ( k epsilon(0,1), ) then the measure of the
( angle C ) is
A.
в.
( c cdot frac{5 pi}{12} )
D.
11
1009 Solve the following equation:
( sin x+sin ^{2} x+cos ^{2} x=0 )
11
1010 Expand ( sin (A+B) ) 11
1011 Change the following degree measures
to radian measure: ( 45^{circ} )
( ^{mathbf{A}} cdot frac{pi}{6} ) radians
B ( cdot frac{pi}{3} ) radians
c. ( frac{pi}{4} ) radians
D ( cdot frac{pi}{2} ) radians
11
1012 If ( tan theta, 2 tan theta+2,3 tan theta+3 ) are in
G.P, then the value of ( frac{7-5 cot theta}{9+4 sqrt{sec ^{2} theta-1}} )
is
A ( cdot frac{12}{5} )
в. ( frac{-33}{28} )
c. ( frac{33}{100} )
D. ( frac{12}{13} )
11
1013 80. In a right angled triangle the hypotenuse is 22 times the
perpendicular drawn from the opposite vertex. Then the
other acute angles of the triangle are
and b. I and
Elt
11
1014 Prove that ( sin 60^{circ} . cos 30^{circ}- )
( cos 60^{circ} cdot sin 30^{circ}=sin 30^{circ} )
11
1015 11. If sin A = sin? B and 2 cos? A = 3 cos? B then the triangle
ABC is
a. right angled
b. obtuse angled
c. isoscelesi
d. equilateral
11
1016 5. Suppose x and y are real numbers such that tan x + tan y=
42 and cotx + coty=49. Then the prime number by which
the value of tan(x + y) is not divisible by is
11
1017 The smallest value of an angle whose
sine is ( -frac{sqrt{mathbf{3}}}{mathbf{2}} ) is
( A cdot 30^{circ} )
B . ( 60^{circ} )
( c cdot 120^{circ} )
D. ( 240^{circ} )
11
1018 Illustration 3.79 In any triangle ABC, prove that
sin A cos(B – C) + sin’B cos(C – A) + sinC cos(A – B)
= 3 sin A sin B sin C
11
1019 If ( boldsymbol{A}+boldsymbol{B}=frac{boldsymbol{pi}}{boldsymbol{4}} ) then value of ( (boldsymbol{1}+ )
( tan A)(1+tan B)= )
( A cdot 4 )
B.
( c cdot 2 )
D. none of these
11
1020 Prove the following identity :
[
frac{1}{sin theta+cos theta}+frac{1}{sin theta-cos theta}=frac{2 sin theta}{1-2 cos ^{2} theta}
]
11
1021 10. The greatest integer less than or equal to –
1
COS 29001
13 sin 250°
11
1022 The values of ( x ) in ( left(0, frac{pi}{2}right) ) satisfying the equation ( sin x cos x=frac{1}{4} ) are
A ( cdot frac{pi}{6}, frac{pi}{12} )
в. ( frac{pi}{12}, frac{5 pi}{12} )
c. ( frac{pi}{8}, frac{3 pi}{8} )
D. ( frac{pi}{8}, frac{pi}{4} )
11
1023 2. Find all the solution of 4 cos²x sin x – 2 sin²x = 3 sinx.
(IIT-JEE 1983)
11
1024 If ( tan x+tan left(x+frac{pi}{3}right)+ )
( tan left(x+frac{2 pi}{3}right)=3, ) then
A ( cdot tan x=1 )
B. ( tan 2 x=1 )
( mathbf{c} cdot tan 3 x=1 )
D. None of these
11
1025 2. a. Draw the graph of y =
(sin x + cos x) from
x=

(IIT-JEE 1979)
b. If cos(a + b) =
€, sin(a – b)
1
, and a, ß lie
, an
between 0 and 77/4, find tan 2 a.
11
1026 If ( I_{n}=int_{0}^{frac{pi}{2}} frac{sin ^{2} n x}{sin ^{2} x} d x, ) then ( I_{1}, I_{2}, I_{2}, cdots )
are in
A. A.P
в. G.
c. н.P.
D. none
11
1027 3. Number of roots of the equation (3 + cos x)2 = 4 –
2 sin®x, x € [0, 51), are
11
1028 28. Prove that a triangle ABC is equilateral if and only if
tan A+tan B+tan C=
(1998 -8 Marks)
11
1029 Illustration 3.19 If tan’A + tanB + tan’C = 3 tan Atan B.
tan C, then prove that triangle ABC is an equilateral triangle.
11
1030 The equation (cos p-1) r4 + (cos p)x+ sin p=0
In the variablex, has real roots. Then p can take any value in
the interval
(1990-2 Marks)
(a) (0,21) (b) (-1,0) ©
(a) (0, 1)
2
2
11
1031 At 4: 24 p.m., how many degrees has the hour hand of a clock moved from its
position at noon?
A ( cdot 135^{circ} )
B. 134
( mathrm{c} cdot 133^{circ} )
D. ( 132^{circ} )
11
1032 Find the value of ( tan left(frac{1}{2} cos ^{-1} frac{sqrt{5}}{3}right) ) 11
1033 4. If sin e- *+ y2 +1
-, then x must be
2x
a. 3d to suono
c. 1
b. -2
d. none of these
11
1034 Solve ( 16^{sin ^{2} x}+16^{cos ^{2} x}=10 ) 11
1035 Simplify, using trigonometric tables
( sin 30^{circ} 30^{prime}+cos 40^{circ} 20^{prime} )
11
1036 30. If tan a is equal to the integral solution of the inequality
4×2 – 16x + 15 < 0 and cos B is equal to the slope of the
bisector of the first quadrant, then sin(a + b) sin(a – b)
is equal to
b.
la viw
11
1037 For all ( theta ) in ( left[0, frac{pi}{2}right] ) Prove that
( cos (sin theta)>sin (cos theta) )
11
1038 If ( boldsymbol{x} cos boldsymbol{theta}=boldsymbol{y} cos left(boldsymbol{theta}+frac{boldsymbol{2} boldsymbol{pi}}{boldsymbol{3}}right)= )
( z cos left(theta+frac{4 pi}{3}right), ) then the value of ( x y+ )
( boldsymbol{y} boldsymbol{z}+boldsymbol{z} boldsymbol{x}= )
( A )
B.
c.
D.
11
1039 20. The range of y such that the equation in x, y + cos x = sin x
has a real solution is
a. [-2, 2]
b. [- V2, V2]
c. [-1, 1]
d. [-1/2, 1/2]
11
1040 Assertion
The equation ( sin x=1, ) has infinite
number of solutions
Reason
The domain of ( f(x)=sin x ) is ( (-infty, infty) )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
1041 9. Number of integral values of a for which the equation
cos²x – sin x + a = 0 has roots when xe (0, 7/2) is
11
1042 25. The value of 2 sin x + tanx is
sin 3x tan 3x
is
11
1043 8. a and ß are the positive acute angles and satisfying
equations 5 sin 2B = 3 sin 2a and tan B = 3 tan a
simultaneously. Then the value of tan a + tan ß is
11
1044 0
HONUT CSO
The number of all possible triplets (a, a, a,) such that
a, + a, cos(2x) +azsin-(x)=0 for all x is (1987-2 Marks)
(a) zero (b) one (c) three
(d) infinite (e) none
16
11
1045 9. The absolute value of the expression tan
1
97
tan —
16
+ tan
137.
– is
16
11
1046 Illustration 3.70 Prove that tan. = 14 + 212 – (V2 + 1) 11
1047 Illustration 2.2
1+sin e
Prove that – = sec 0 + tan e.
V1–sino
11
1048 12. If x = sec
– tan o and y = cosec 0 + cot 0, then
y +1
y-1
a.
X=

=
y-1
b.
x=
y +1
um
c.
y =
1 + x
1-x
d. xy + x – y + 1 = 0
11
1049 ( i f angle A=angle B=45^{circ} ) verify that ( , sin )
( (A+B)=sin A cos B+cos A sin B )
11
1050 56. Find in radians the angle between
the hour hand and the minute
hand of a clock at half past three.
radians
radians
radians
radians
radians
11
1051 Find the value of the other five
trigonometric functions for the
following:
( tan x=frac{3}{4}, x ) in quadrant ( I I I )
11
1052 The value of
1) ( sin left(4 theta+frac{pi}{2}right) )
2) ( tan theta+sec theta )
11
1053 If ( sin alpha+sin beta=a ) and ( cos alpha-cos beta= )
( b, ) then ( tan frac{alpha-beta}{2} ) is equal to
A. ( -frac{a}{b} )
в. ( -frac{b}{a} )
c. ( sqrt{a^{2}-b^{2}} )
D. None of these
11
1054 Find the value of
( frac{sqrt{1+sin 2 A}+sqrt{1-sin 2 A}}{sqrt{1+sin 2 A}-sqrt{1-sin 2 A}} ) when
( |tan A|<1 ) and ( |A| ) is acute
( mathbf{A} cdot tan A )
B. ( -tan A )
( c cdot cot A )
D. – cot ( A )
11
1055 State whether true or false:
( sin 2 x+2 sin 4 x+sin 6 x= )
( 4 cos ^{2} x sin 4 x )
A. True
B. False
11
1056 8.
Find all the solution of 4 cosxsin x – 2 sina x = 3 sin x
(1983 – 2 Marks)
11
1057 Find the number of values of ( boldsymbol{theta} )
satisfying the equation ( sin 3 theta= ) ( 4 sin theta cdot sin 2 theta cdot sin 4 theta ) in ( 0 leq theta leq 2 pi )
11
1058 If ( 4 cos ^{2} theta+15 cos theta-4=0, ) find the
value of ( left(log _{2} cos thetaright)^{-5}: )
A . -1
в. ( -frac{1}{32} )
( c . )
D. none of these
11
1059 Evaluate
( sin 1^{0} sin 2^{0} sin 3^{0} ldots sin 179^{0} sin 180^{0} )
11
1060 Illustration 4.19 Solve V5 – 2 sin x = 6 sin x-1 11
1061 Solve ( sin ^{2} theta-cos theta=frac{1}{4} .0 leq theta leq 2 pi ) 11
1062 75. If sin* 0 + cos* 0 = 2 sina o
cos2 O, O is an acute angle, then
the value of tan 0 is
(2) 2
(3) 2 (4) O
(1) 1
11
1063 If ( tan theta=frac{-4}{3} ) then ( sin theta ) is
A ( cdot frac{-4}{5} ) but not ( frac{-4}{3} )
B. ( frac{-4}{5} ) or ( frac{4}{5} )
c. ( frac{4}{3} ) but not ( frac{-4}{5} )
D. None of thses.
11
1064 Area of circle in which a chord of length
( 2 sqrt{3} ) units, subtends angle ( 120^{circ} ) at its
center is-
A . ( pi ) sq units
B. 2 ( pi ) sq units
c. ( 4 pi ) sq units
D. None of these
11
1065 If ( sec A+tan A=m ) and ( sec A- )
( tan A=n, ) find the value of ( sqrt{m n} )
( A cdot 0 )
B. ±1
( c .pm 2 )
( D ldots 3 )
11
1066 If ( sin x+sin ^{2} x=1 ) then the value of
( cos ^{2} x+cos ^{4} x ) is equal to
( A )
B. ( frac{1}{2} )
c. ( frac{1}{3 sqrt{3}} )
D. ( frac{3 sqrt{5}-5}{2} )
11
1067 Illustration 3.69 Show that
4 sin 27° = (5+15)1/2 – (3 – 75)12.
11
1068 Prove ( operatorname{cosec}^{6} boldsymbol{A}-cos ^{6} boldsymbol{A}= )
( mathbf{3} cot ^{2} boldsymbol{A} operatorname{cosec}^{2} boldsymbol{A}+mathbf{1} )
11
1069 Illustration 3.104 Prove that in a A ABC, sin A + sinB +
sin
sin? Csı
11
1070 Find the values of the following:
( sin 120^{circ} )
11
1071 Illustration 4.7 Find general value of Owhich satisfies both
sin 0 = -1/2 and tan 0 = 1/73 simultaneously.
7
11
1072 19. If sin’x cos 3x + cos’x sin 3x = 3/8, then the value of
8sin 4x is
19. If sin x cos 3x + cos’r sin 3x = 318, then the value of
11
1073 28.
Let S =
x e(-TT,t):
X
0
,-
*. The sum of all distinct
solutions of the equation 13 sec x + cosec x + 2(tan x –
cot x)=0 in the set S is equal to (JEE Adv. 2016)
(b)
276
(©) 0
11
1074 Illustration 4.28
Solve cos x cos 2x cos 3x = 1/4.
11
1075 If ( tan ^{4} theta+cot ^{4} theta=A, ) then
A ( . A>2 )
в. ( A geq 2 )
c. ( A>4 )
D. ( A geq 4 )
11
1076 Solve the following equation:
( 2^{sin ^{2} x}+2^{cos ^{2} x}=2 sqrt{2} )
11
1077 Evaluate
( frac{sin ^{2} 63^{circ}+sin ^{2} 27^{circ}}{cos ^{2} 17^{circ}+cos ^{2} 73^{circ}} )
2) ( sin 25^{circ} cos 65^{circ}+cos 25^{circ} sin 65^{circ} )
11
1078 If ( sin alpha sin beta-cos alpha cos beta+1=0 ), then
show that ( 1+cot alpha tan beta=0 )
11
1079 ( 1+cos 56^{circ}+cos 58^{circ}-cos 66^{circ}= )
( m cos 28^{circ} cos 29^{circ} sin 33^{circ} . ) Find ( m )
11
1080 3. A general solution of tan+ cos20 = 1 is (n € Z)
a. n-
b. 211
+
c. nn +
d. na
11
1081 Illustration 3.72
Prove that sin 0 + sin 30 + sin 50 + …
sin?ne
sin e
+ sin(2n – 1)
=
11
1082 Find the general solution of the
equation ( sin 2 x+sin 4 x+sin 6 x=0 )
11
1083 18. The smallest positive value of x (in radians) satisfying the
equation logcosx
SIN X
= 2 – logsec x(tan x), is
11
1084 Expand
( cos theta+cos phi= )
11
1085 If ( boldsymbol{theta} inleft(mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right), ) then the value of
( cos left(theta-frac{pi}{4}right) ) lies in the interval
( ^{mathbf{A}} cdotleft(frac{1}{2}, 1right) )
B ( cdotleft(frac{1}{sqrt{2}}, 1right) )
( ^{mathbf{c}} cdotleft(-frac{1}{sqrt{2}}, 1right) )
D. (0,1)
11
1086 Prove that ( cos ^{2} A+cos ^{2}(A+120)+ )
( cos ^{2}(A-120)=frac{3}{2} )
11
1087 The total no. of solution of equation ( |cot x|=cot x+frac{1}{sin x}, x in[0,3 pi] ) is
equal to
( A cdot 3 )
B . 2
( c cdot 1 )
( D )
11
1088 21. In triangle ABC, tan (A – B) + tan (B-C)+ tan (C-A)=0.
Prove that the triangle isisosceles.
11
1089 ( cot A ) is the product of ( cot ) and ( A )
A. True
B. False
11
1090 ( boldsymbol{I} boldsymbol{f} sin 5 boldsymbol{theta}=boldsymbol{a} sin ^{5} boldsymbol{theta}+boldsymbol{b} sin ^{3} boldsymbol{theta}+ )
( operatorname{csin} theta ) then
( (A) a-2 c=5 )
(B) ( a-2 c=6 )
(C) ( b+3 c=5 )
(D) ( b+3 c=-5 )
The general solution of ( tan 3 theta tan theta= )
1 is given ( (n in boldsymbol{I}) )
( mathbf{A} cdot a=20, b=-10, c=5 )
B . ( a=1, b=-20, c=5 )
( mathbf{c} . a=16, b=-2, c=5 )
D. ( a=16, b=-20, c=5 )
11
1091 Find the measure of an angle in degrees formed by an arc of ( 2.5 mathrm{cm} ) length at the centre of a circle with ( 15 mathrm{cm} ) radius 11
1092 General solution for ( |sin x|=cos x ) is
A ( cdot 2 n pi+frac{pi}{4}, n in I )
В ( cdot 2 n pi pm frac{pi}{4}, n in I )
c. ( n pi+frac{pi}{4}, n in I )
D. None of these
11
1093 Given that ( A ) is positive acute angle and ( sin A=frac{sqrt{3}-1}{2}, ) then ( A ) take the value
( (s)- )
A .15
B. 30
c. 45
D. 75
11
1094 3. The general solution of the equation sin x – 3 sin 2x +
sin 3x = cos x – 3 cos 2x + cos 3x is (ne 2)
a. nt+
ь пли
та
28
C. (+1)» MT
d. 2nd+ cos!?
11
1095 General value of ( theta ) satisfying equation
( tan ^{2} theta+sec 2 theta=1 ) is
A ( . n pi )
B . ( n pi+frac{pi}{3} )
( mathbf{c} cdot n pi-frac{pi}{3} )
D. All of these
11
1096 19. The maximum value of (cosa,).(COS C.,)…(cosa), under
the restrictions
05@,, ,, …,2., and (cot (,).(cot ay)… (cot Qn)= 1 is
(20015)
(a) 1/22 (6) 1/2 © 1/2n di
11
1097 12. Let k be sum of all x in the interval [0, 21) such that
3cot-x + 8cot x + 3 = 0, then the value of k/a is
11
1098 If
( [x] ) denotes the greatest integer ( leq x )
then the system of linear equations ( [sin theta] x+[-cos theta] y=0[cot theta] x+y=0 )
A ( cdot ) Have infinitely many solutions if ( theta epsilonleft(frac{pi}{2}, frac{2 pi}{3}right) cup ) ( left(pi, frac{7 pi}{6}right) )
B. Have infinitely many solutions if ( theta epsilonleft(frac{pi}{2}, frac{2 pi}{3}right) ) and has unique solution if ( theta epsilonleft(pi, frac{7 pi}{6}right) )
c. Has a unique solution if ( theta epsilonleft(frac{pi}{2}, frac{2 pi}{3}right) ) and have nfinitely many solutions if ( theta epsilonleft(pi, frac{7 pi}{6}right) )
D. Has a unique solution if ( theta epsilonleft(frac{pi}{2}, frac{2 pi}{3}right) cupleft(pi, frac{7 pi}{6}right) )
11
1099 14. If (1 + tan a) (1 + tan 4a) = 2, a E (0, 1/16), then a is
..
mequal to
it to usd
11
1100 Illustration 3.62
If x + y + z = xyz, prove that
2x
I 2x
2y
2z.
1-r2
1- x² 1 – 2 1 – 2²
परx
22 x
11
1101 Express the sexagesimal measure ( 15^{circ} )
as radian measure
11
1102 (sinx + cosx) from x-
2.
(a) Draw the graph of y=
(6) If cos (a + b) = 5, sin (Q – B) = g, and a, lies
between 0 and -, find tan2a.
(1979)
11
1103 The values of ( x epsilon[-2 pi, 2 pi] ) such that
( frac{sin x+i cos x}{1+i}, i=sqrt{-1}, ) is purely
imaginary, are given by
A ( cdot n pi-frac{pi}{4} )
в. ( n pi+frac{pi}{4} )
( c cdot n pi )
D. none of these
11
1104 24.
The values of 0 € (0,210) for which 2 sin20-5 sino +2>0
are
(2006 – 3M, -1)
48
11
1105 91. In triangle ABC, tan A + tan B + tan C = 6 and tan A tan B
= 2, then the values of tan A, tan B, tan C are, respectively
soba. 1, 2, 3 tab. 3, 2/3, 7/3
c. 4, 1/2, 3/2
d. none of these
11
1106 The range of ( boldsymbol{f}(boldsymbol{x})= )
( frac{1}{5 sin x-6} epsilon[-a,-1 / b] ) Find ( a+b )
11
1107 2. If A = sin 45° + cos 45º and B = sin 44° + cos 44°, then
a. A > B
b. A<B
c. A=B
d. none of these
11
1108 The number of solutions of ( cos x= )
( |1+sin x|, 0 leq x leq 3 pi, ) is
A. 3
B. 2
( c cdot 4 )
D. none of these
11
1109 25. For 0 < x, y < t, the number of ordered pairs (x, y)
satisfying the system of equations cot?(x – y) –
(1+73)cot(x – y) + V3 = 0 and cos y =
a. 0
c. 2
d. 3
b. 1 b
otto
11
1110 67. If sec 0 = cosec o, where and
are acute angles, then the value
of cosec (0 + 0) is
(1) 1
(2) O
(3) undefined (4) 12
11
1111 Let ( x=sin 1^{circ}, ) then the value of the
( operatorname{expression} frac{1}{cos 0^{circ} cdot cos 1^{circ}}+ )
( frac{1}{cos 1^{circ} cdot cos 2^{circ}}+frac{1}{cos 2^{circ} cdot cos 3^{circ}}+dots+ )
( frac{1}{cos 44^{circ} cdot cos 45^{circ}} ) is equal to
A . ( x )
в. ( frac{1}{x} )
( c cdot frac{sqrt{2}}{x} )
D. ( frac{x}{sqrt{2}} )
11
1112 A flag-staff 20 metres long standing on a wall 10 metres high subtends an
angle whose tangent is 0.5 at a point on
the ground. If ( theta ) is the angle subtended
by wall at that point then ( tan theta= )
A . 1
B. ( frac{1}{3} )
c. ( _{1 text { or }} frac{1}{3} )
D.
11
1113 U
IL .
Illustration 4.53 Solve sin’x + cos²y= 2sec z for x, y, and z.
11
1114 tanº e
11. If sec o
+

7, then prove that |b|slal.
b
a+b
11
1115 Illustration 4.50 Find the number of solutions of sin’xcos²x
= 1 + cos² x sinºx in the interval [0, 211].
11
1116 Simplify the following expression:
( frac{1+sin +2 x}{(sin x+cos x)^{2}} )
11
1117 Evaluate ( : frac{operatorname{cosec}^{2} mathbf{6 3}^{mathbf{0}}+tan ^{2} mathbf{2 4}^{mathbf{0}}}{cot ^{2} mathbf{6 6}^{mathbf{0}}+mathbf{s e c}^{mathbf{2}} mathbf{2 7}^{mathbf{0}}}+ )
( frac{mathbf{s i n}^{2} mathbf{6 3}^{mathbf{0}}+cos mathbf{6 3}^{mathbf{0}} sin mathbf{2 7}^{mathbf{0}}+sin mathbf{2 7}^{mathbf{0}} mathbf{s e c 6}}{mathbf{2}left(operatorname{cosec}^{mathbf{2}} mathbf{6 5}^{mathbf{0}}-mathbf{t a n}^{mathbf{2}} mathbf{2 5}^{mathbf{0}}right)} )
11
1118 Transform the following expression
( frac{tan alpha+tan beta}{cot alpha+cot beta}+[cos (alpha- )
( boldsymbol{beta}) boldsymbol{s e c}(boldsymbol{alpha}+boldsymbol{beta})+mathbf{1}]^{-1} )
11
1119 ( cot theta+tan theta=x ) and ( sec theta-cos theta= )
( y ) then ( left(x^{2} yright)^{2 / 3}-left(x y^{2}right)^{2 / 3} )
11
1120 2. If ABC is a triangle and tan
tan
pla
, tan
are in H.P.
then find the minimum value of cot B/2.
11
1121 2.
Ifa+B+ y2, then
(1979)
(©) tan + tan + tan — tan tan tanz
tan
(d) None of these
11
1122 Illustration 4.37
Solve 2 sinºx — 5 sinx cos x – 8 cos x=-2.
11
1123 29. The value of
29. The value of
( m
is equal
(k-1)
(-0)-G
sin
+
}.
sin
a ka
– +
A
6
to
(a) 3-13
© 2(73-1)
(JEE Adv. 2016)
(b) 2(3-13)
(d) 2(2-13)
11
1124 93. If a sin x + b cos(x + 0) + b cos(x – 0) = d, then the
minimum value of cose is equal to
2.
2161
zla
c.
va? – 4
d. none of these
d. none of these
2 dl
11
1125 Solve ( cos (sin theta)=sin (cos theta) ) 11
1126 Prove that:-
( sin 20 sin 40 sin 60 sin 80=frac{3}{16} )
11
1127 ( cos frac{2 pi}{15} cos frac{4 pi}{15} cos frac{8 pi}{15} cos frac{14 pi}{15} )
A . ( 1 / 4 )
в. ( 1 / 8 )
c. ( 1 / 16 )
D. ( 1 / 32 )
11
1128 Illustration 4.17
Solve 4 cos 0 – 3 sec 0 = tan 0.
11
1129 The general solution of the equation ( sin ^{100} x-cos ^{100} x=1 ) is
A ( cdot 2 n pi+frac{pi}{3}, n epsilon I )
В ( cdot n pi+frac{pi}{2}, n epsilon I )
c. ( n pi+frac{pi}{4}, n epsilon I )
D. ( 2 n pi-frac{pi}{3}, n epsilon I )
11
1130 63. If 9 sino + 40 cos0 = 41, then
coso will be
es
BA TL
11
1131 57. If , and O2 are two values lying in [0, 21] for which
tan 0= 2, then tan tanz is equal to
inupa ar a.
0
2 03 O b. -1 209 to Sleva
c. 2
d. 1
11
1132 One angle of a triangle is ( frac{2 x}{3} ) grad, another is ( frac{3 x}{2} ) degrees, whilst the third is ( frac{pi x}{75} ) radians. Express all angles in
degress.
A. Hence three angles of the triangle are ( 43^{circ}, 30^{circ}, 30^{circ} )
B. Hence three angles of the triangle are ( 24^{circ}, 60^{circ}, 96^{circ} )
C. Hence three angles of the triangle are ( 74^{circ}, 27^{circ}, 98^{circ} )
D. Hence three angles of the triangle are ( 30^{circ}, 60^{circ}, 90^{circ} )
11
1133 8. If a + B = /2 and 3+ y=ą, then tan a equals
a. 2 (tan B+ tan b. tanß+ tan y
c. tan ß + 2 tan y d. 2 tan ß+ tan y
11
1134 51. If sin x + cosec x + tan y+cot y=4 where x and y el 0,
then tán is a root of the equation
a. O2 + 2a + 1 = 0 с. 02 + 2a – 1 = 0
c. 20-2a-1=0 d. o – a- 1 = 0
11
1135 If ( tan 2 A=cot (A-18), ) where ( 2 A ) is an
acute angle, then find the value of ( boldsymbol{A} )
11
1136 The value of
( frac{tan ^{2} 60^{0}-2 tan ^{2} 45^{0}+sec ^{2} 30^{0}}{3 sin ^{2} 45^{0} sin 90^{0}+cos ^{2} 60^{0} cos ^{3} 0^{0}} )
A ( cdot frac{49}{12} )
B. ( frac{7}{3} )
c. ( frac{14}{9} )
D.
11
1137 Illustration 3.70
Prove that tan—
16
11
1138 If ( sqrt{3} tan 2 theta+sqrt{3} tan 3 theta+ )
( tan 2 theta tan 3 theta=1, ) then the general
value of ( theta ) is
A ( . quad n pi+frac{pi}{5} )
B ( cdotleft(n+frac{1}{6}right) frac{pi}{5} )
( ^{c} cdotleft(2 n pm frac{1}{6}right) frac{pi}{5} )
D. ( left(n+frac{1}{3}right) frac{pi}{5} )
11
1139 tan
17. The number of distinct real roots of the equation
tan — 21x_=-5
x²+x+1
b. 5
c. 6
d. none of these
2
a. 4
11
1140 Illustration 4.57
Solve the equation
cos (sin x + V2 cosro)- tan” x + ” tan?x) = 1.
11
1141 Illustration 3.76 Prove that in triangle ABC, cos? A + cos²B
-cosC = 1-2 sin A sin B cos C.
11
1142 If ( sec theta+tan theta=k, cos theta= )
A. ( frac{k^{2}+1}{2 k} )
в. ( frac{2 k}{k^{2}+1} )
c. ( frac{k}{k^{2}+1} )
D. ( frac{k}{k^{2}-1} )
11
1143 The value of ( cos left(36^{circ}-Aright) cos left(36^{circ}+right. )
( A)+cos left(54^{circ}+Aright) cos left(54^{circ}-Aright) ) is?
( mathbf{A} cdot sin 2 A )
B. ( cos 2 A )
( c cdot cos 3 A )
( mathbf{D} cdot sin 3 A )
11
1144 20. The value of
cos-10° – cos 10° cos50° + cos250° is:
JEEM 2019-9 April (M
(b) 3/4
(a
©
+ cos200
(1 + cos20°)
(d) 312
11
1145 Ma Tuusu-
2. If 2 cos x + sin x = 1, then find the value of 7 cos x + 6 sinx.
2
2.
11
1146 3. Find the values of x and y for which cosec 0=
satisfied.
X2 is
11
1147 Illustration 4.47 Find the smallest positive values of x and
y satisfying x – y = – and cotx + coty= 2.
11
1148 If ( boldsymbol{x}=tan boldsymbol{theta}+cot boldsymbol{theta}, boldsymbol{y}=cos boldsymbol{theta}-sin theta )
then
A. ( x=y )
B. ( frac{1-y^{2}}{2}=frac{1}{x} )
( ^{mathbf{c}} cdot frac{y^{2}-1}{2}=frac{1}{x} )
D. ( frac{1+y^{2}}{2}=frac{1}{x} )
11
1149 Show that
( tan 3 x tan 2 x tan x=tan 3 x- )
( tan 2 x-tan x )
11
1150 ( frac{sin A}{1+cos A}+frac{sin A}{1-cos A} ) is equal to
( A cdot sin A )
B. ( 2 operatorname{cosec} A )
( c cdot cos A )
D. None of these
11
1151 100. If x sin a + y sin 2a + z sin 3a = sin 4a,
x sin b + y sin 2b + z sin 3b = sin 4b,
x sin c + y sin 2c + z sin 3c = sin 4c,
then the roots of the equation –
(z-x
= 0, a, b, c nt, are
T 8
a. sin a, sin b, sinc b. cos a, cos b, cos c
c. sin 2a, sin 2b, sin 2c d. cos 2a, cos 26 cos 2c
11
1152 Evaluate ( frac{sin ^{2} 63^{circ}+sin ^{2} 27^{circ}}{cos ^{2} 17^{circ}+cos ^{2} 73^{circ}} ) 11
1153 If ( 15 tan ^{2} theta+4 sec ^{2} theta=23 ) then
( tan ^{2} theta=dots )
A ( cdot frac{27}{15} )
B. 45
( c cdot frac{19}{11} )
D.
11
1154 Illustration 4.63
< 370/2.
Solve 2 cos²0 + sin 0 < 2, where it/2 s 0
11
1155 Solve :-
[
sin 45^{circ}+cos 45^{circ}
]
11
1156 21.
The number of integral values of k for which the equation 7
cos x +5 sin x =2k + 1 has a solution is
(2002)
(a) 4 (b) 8 (c) 10 (d) 12
11
1157 ( operatorname{Given} frac{x-x tan ^{2} 30^{circ}}{1+tan ^{2} 30^{circ}}=sin ^{2} 30^{circ}+ )
( 4 cot ^{2} 45^{circ}-sec ^{2} 45^{circ} . ) Then the value of
( boldsymbol{x} )
( A cdot frac{3}{2} )
в. ( frac{5}{2} )
( c cdot frac{9}{2} )
D. none of these
11
1158 If ( boldsymbol{A}+boldsymbol{B}=frac{boldsymbol{pi}}{mathbf{3}} ) and ( cos boldsymbol{A}+cos boldsymbol{B}=mathbf{1} )
then which of the following are true:
This question has multiple correct options
A ( cdot cos (A-B)=frac{1}{3} )
B. ( cos (A-B)=-frac{1}{3} )
C ( cdot|cos A-cos B|=sqrt{2 / 3} )
D. ( |cos A-cos B|=frac{1}{sqrt{3}} )
11
1159 13. Solve the following system of simultaneous equations for
x and y:
4sin x + 31/cosy = 11
5 x 16sin x – 2 x 31/cosy = 2
11
1160 60. If 0° <e < 90°, the value of
sin e + cos O is
(1) equal to 1
(2) greater than 1
(3) less than 1
(4) equal to 2
11
1161 For ( boldsymbol{x} in(mathbf{0}, boldsymbol{pi}), ) the equation ( sin boldsymbol{x}+ )
( 2 sin 2 x-sin 3 x=3, ) has
A. infinitely many solutions
B. three solutions
c. one solution
D. no solution
11
1162 Illustration 3.71 Find the value of cos
+ cos
11
1163 A unit radian is approximately equal to
A ( cdot 57^{circ} 17^{prime} 43^{prime} )
,
B . ( 57^{circ} 17^{prime} 45 ” )
c. ( 57^{circ} ) 17′ ( 47 ” )
D. ( 57^{circ} 17^{prime} 49^{prime} )
11
1164 Solve the following equation:
( cos x cos 2 x cos 3 x=frac{1}{4} )
11
1165 4. Solve sinx + siny = sin(x + y) and [xl + byl = 1.
44
11
1166 ( cot B=2 tan (A-B)=>2 tan B+ )
( cot B ) is equal to
( mathbf{A} cdot tan A )
B. ( cot A )
( c cdot 2 tan A )
D. ( 2 cot A )
11
1167 Points in which abscissa and ordinate
have different signs will lie in
A. Ist and IIIrd quadrants
B. Illrd and IVth quadrants
c. Ilnd and Illrd quadrants
D. IInd and IVth quadrants
11
1168 5
Tet a and ß be any two positive values of x for which
nost. I cos x , and 1 – 3 cos x are in G.P. The minimum
value of a-Bis
d. none of these
11
1169 4. Number of solution(s) of the equation
sin
x
cos 3x
sin 3x
cos 9x
+ cip- in the interval ( 0. A) is
sin 9x
-= 0 in the interval
cos 27 x
11
1170 Find the angle measure of 4 radians.
A ( cdot 114.591^{circ} )
В. ( 141.372^{circ} )
c. ( 229.183^{circ} )
D . ( 282.743^{circ} )
E ( .458 .366^{circ} )
11
1171 Evaluate:
( frac{sec x^{o}+tan x^{o}}{sec x^{o}-tan x^{o}} )
11
1172 Illustration 4.5 Find the values of which satisfy r sin 0=3
and r=4 (1 + sin 8), 0 <O<2n.
11
1173 If ( sin theta=cos left(2 theta-45^{circ}right), quad 0<(2 theta- )
( left.45^{circ}right)<90^{circ}, ) then ( tan theta )
A . -1
B.
c.
D. ( frac{1}{sqrt{3}} )
11
1174 ( frac{tan theta+sec theta-1}{tan theta-sec theta+1}= )
( mathbf{A} cdot frac{cos theta}{1+sin theta} )
B. ( frac{1+cos theta}{sin theta} )
( mathbf{C} cdot frac{1+sin theta}{cos theta} )
D. ( frac{1-sin theta}{cos theta} )
11
1175 7. Let f(0) = sin 0 (sin 0+ sin 38). Then f (O) is
a. 20 only when 020 b. S 0 for all real 0
c. 20 for all real 0 d. <0 only when oso
11
1176 Find the value of
( left(tan 2^{circ} tan 4^{circ} tan 6^{circ}——tan 8right. )
11
1177 The angle of the sun above the horizon
is 27.5 degrees. Find the approximate length of the shadow of a person who is
4.75 feet tall.
A . 4.75
в. 2.47
c. 4.65
D. 9.12
E . 4.86
11
1178 Prove that ( 2 sec ^{2} theta-sec ^{4} theta- )
( 2 cos e c^{2} theta+cos e c^{4} theta=cot ^{4} theta-tan ^{4} theta )
11
1179 If ( 6 sin ^{2} theta-sin theta=1 ) and ( 0 leq theta leq pi )
calculate the value of ( sin theta )
A ( cdot frac{1}{6} )
B.
( c cdot frac{1}{2} )
D. 19
E . 30
11
1180 64. The general solution of the equation sin100x – cos100x = 1 is
a. 2nd+ , nel b. nt+ – ,ne I
c. nn +
-,nel
ne
d. 2nd – – ,ne I
11
1181 Find the solution of
(i) ( 10 cos ^{3} x-16 cos x=3 cos 2 x+3, ) in
the interval ( [-boldsymbol{pi}, boldsymbol{pi}] )
( (i i) 2(cos x+cos 2 x)+sin 2 x(1+ )
( 2 cos x)=2 sin x ) in the interval ( [-pi, pi] )
11
1182 General value of ( theta ) satisfying the
eqation ( tan ^{2} theta+sec 2 theta=1 ) is
( mathbf{A} cdot m pi, n pi pm frac{pi}{3}, m, n in I )
В ( cdot m pi, n pi pm frac{pi}{4}, m, n in I )
c. ( m pi, n pi pm frac{pi}{6}, m, n in I )
D. ( m pi, n pi pm frac{pi}{8}, m, n in I )
11
1183 Illustration 3.31 Prove that
sin 5A – sin 3A
= tan A
cos 5A + cos 3A
sin A + sin 3A
-= tan 2A
cos A + cos 3A
11
1184 82. The value of the following is:
(tan 20°)2 (cot 20°)2
(cosec 70°)2 + (sec 70°)2
2tan 15°. tan 45º tan 75°
(1) 1
(2) 4
(3) 3
(4) 2
11
1185 6. Number of integral value(s) of m for which the equation
4m – 6
sin x – 13 cos x = has solutions, x = [0, 21), is
4 – m
11
1186 Solve :
[
tan ^{2} theta-2 sin theta=0
]
11
1187 If ( tan B=frac{2 sin A sin C}{sin (A+C)} ) then
( tan A, tan B, tan C ) are in
A . A.
в. G.
c. н.
D. AGP
11
1188 Illustration 4.61 If m and n (n > m) are positive integers,
then find the number of solutions of the equation n|sin x1 =
m|cos x| for x E[0, 21). Also find the solution.
11
1189 ( sec theta-tan theta=6 )
Then ( sec theta+tan theta=? )
11
1190 Verify that:
( cos 60^{circ}=frac{1-tan ^{2} 30^{circ}}{1+tan ^{2} 30^{circ}}=frac{1}{2} )
11
1191 The radius of circle is ( 9 mathrm{cm} ). Find the
length of an arc of this circle which cuts
off a chord of length equal to the radius.
11
1192 Find the value of
( sin 135^{circ} )
11
1193 Illustration 4.35
Solve tan 0 + tan 20 + V3 tan 0 tan 20 =
V3
11
1194 Prove ( : frac{cos ^{2} theta}{1-tan theta}+frac{sin ^{3} theta}{sin theta-cos theta}=1+ )
( sin theta cos theta )
11
1195 8. The expression (tan^x + 2 tan²x + 1) cos²x, when x=īt/12,
can be equal to
a. 4(2 – 13)
c. 16 cos? tt/12 d. 16 sin®īt/
12 0
can be equal
b, 46 13
b. 4( N2 +1)
11
1196 1. If 4 sin4x + cos*x = 1, then x is equal to (n e 2)
a. nt
b. nnt
sin-1
2na
d. 2nd =
11
1197 11. Eliminate x from equations sin(a + x) = 2b and sin(a – x)
= 2c.
11
1198 If ( sec theta=frac{25}{7}, ) then find the value of
( tan theta ? )
11
1199 9. Solve tan ( coso – cot ( sin o) 11
1200 In ( triangle A B C, ) if ( a, b, c ) are in ( A . P . ) then ( cot frac{boldsymbol{A}}{2} cot frac{boldsymbol{C}}{2}= )
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot 3 )
D.
11
1201 10. The value of 0 € (0,21) for which 2 sine – 5 sin 0+2>
O is
417
(IIT-JEE 2006
11
1202 If ( 0<x leq frac{pi}{2}, ) then ( sin x+operatorname{cosec} x geq )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
11
1203 Let ( X ) be the solution set of the equation ( boldsymbol{A}^{x}=boldsymbol{I}, ) where ( boldsymbol{A}=left[begin{array}{ccc}mathbf{0} & mathbf{1} & mathbf{- 1} \ mathbf{4} & mathbf{- 3} & mathbf{4} \ mathbf{3} & mathbf{- 3} & mathbf{4}end{array}right] ) and
is the corresponding unit matrix and
( boldsymbol{x} subseteq boldsymbol{N}, ) then the minimum value of
( sumleft(cos ^{x} theta+sin ^{x} thetaright), theta in R )
11
1204 f ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=boldsymbol{x} boldsymbol{y} boldsymbol{z}, ) prove that
( frac{2 x}{1-x^{2}}+frac{2 y}{1-y^{2}}+frac{2 z}{1-z^{2}}= )
( frac{2 x}{1-x^{2}} frac{2 y}{1-y^{2}} frac{2 z}{1-z^{2}} )
11
1205 If ( boldsymbol{A}-boldsymbol{B}=boldsymbol{C} ) and ( boldsymbol{A}+boldsymbol{B}=frac{pi}{2} ) then
( tan A=tan B+2 tan C )
11
1206 +1+1+1 is
+-
6. The value of – -+ –
tan a tan ß
tand
a. – 8
c. 2/3
tan y
b. 8
d. 1/3
11
1207 92. If cos x + cos y – cos(x + y) =
=
=
2
th
a. x + y = 0
c. x=y
b. x = 2y
d. 2x = y
11
1208 Find the general solution of the equation ( tan ^{2} alpha+2 sqrt{3} tan alpha=1 ) 11
1209 28. Iff(x) = cos_0+ sec 6, then
a. f(x) f(x) > 1
d. f(x) 22
11
1210 Tllustration 3.64 Prove that tan — is a root of polynomial
equation 5×4 – 10x² + 1 = 0.
10
11
1211 Solve the following equation:
( tan x=-1 )
11
1212 Express the following angle in degree ( left(-frac{7 pi}{12}right)^{c} ) 11
1213 45. If a, b, y, are the smallest positive angles in ascending
order of magnitude which have their sines equal to the
positive quantity k, then the value of 4 sin + 3 sin Þ
+2 sin
+ sin
is equal to
2
b. 2/1+k
a. 2/1-k
a Vith
d. none of these
11
1214 If ( boldsymbol{alpha} ) and ( beta ) are two different values of ( boldsymbol{theta} )
lying between 0 and ( 2 pi ) which satisfy
( 3 cos theta+4 sin theta=6 . ) Find the value of
( sin (alpha+beta) )
11
1215 16. Let f (x) = ab sin x + b V1-a² cos x+c, where |al 0 then
a. maximum value of f(x) is b if c = 0
b. difference of maximum and minimum values of f(x)
is 2b
g
o
c. f(x) = c if x = – cos-1
a
d. f(x) = c if x = cos’ a
11
1216 Illustration 3.48 If p(x) = sin x (sinºx+3)+cosx (cos’x +4)
+ (1/2) sin 2x + 5, then find the range of p(x).
11
1217 The area of the circle is ( 25 pi ) sq. cms.
Find the length of its arc subtending an angle of ( 144^{circ} ) at the centre. Also find the
area of the corresponding sector.
11
1218 Find the number of solutions of the
equations;
( |cot x|=cot x+frac{1}{sin x}, ) where ( x in )
( [0,2 pi] )
11
1219 ( fleft(m^{2} cos frac{2 pi}{15} cos frac{4 pi}{15} cos frac{8 pi}{15} cos frac{14 pi}{15}=n^{2}right. )
then find the value of ( frac{m^{2}-n^{2}}{n^{2}} )
11
1220 Evaluate :
( frac{2 cos 67^{circ}}{sin 23^{circ}}-frac{tan 20^{circ}}{cot 50^{circ}}-cos 0^{circ} )
11
1221 Solve:
( frac{x sin x}{1+cos x} )
11
1222 Solve:
( 3 sin ^{2} x-7 sin x+2=0 )
11
1223 A flag-staff stands on a tower which is on level ground. The total height of the flag-staff and tower taken together is
300 metres. The flag-staff subtends an angle of ( tan ^{-1}left(frac{1}{5}right) ) at a point ( P ) on the level ground at a distance 300 metres from the foot of the tower. The height of the tower is:
A. 100 metres
B. 200 metres
c. 250 metres
D. 300 metres
11
1224 The value of ( sin 51^{circ}+cos 81^{circ} ) is
A . ( cos 21^{circ} )
B. ( sin 21^{circ} )
( mathbf{c} cdot cos 42^{circ} )
( mathbf{D} cdot sin 42^{circ} )
11
1225 9.
Let A and B denote the statements
A: cos a + cos B + cos y=0
B: sin a + sin ß+sin y=0
If cos (B-Y)+cos (y-a)+cos (a-B)= , then : [2009
(a) A is false and B is true (b) both A and B are true
(c) both A and B are false (d) A is true and B is false
11
1226 Illustration 3.80 If A + B + C = Tt, prove that cot A + cot B
+ cot C – cosec A. cosec B. cosec C = cot A . cot B . cot C.
11
1227 14. The number of solutions of equation 6 cos 20+2 cos?(@/2)
+ 2 sin?O=0,-< < it is
a. 3
b. 4
c. 5
d. 6
11
1228 Illustration 3.74 Prove that 2 sin 2° + 4 sin 4° + 6 sin 6° +
… + 180 sin 180º = 90 cot 10°.
Ilustration 3: 75 prove that 2 sin 2° + sin 4 + 6 sin 6° +
11
1229 10. The value of (tan? “* + tan ? 2 + tan? 34) (cov? ”
+ cot2 217 + cot? 37 is
a. 105
c. 210
b. 35
d. none of these
11
1230 Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, shall be three times as old as you will be.” Is not this interesting? Represent this situation algebraically. 11
1231 ( sin 1^{circ} sin 2^{circ} sin 3^{circ} dots dots sin 180^{circ} ) is equal
to
( mathbf{A} cdot mathbf{1} )
B. –
c. 0
D. ( frac{1}{2} )
11
1232 14. If sinx – a sin x + b = 0 has only one solution in (0,7),
then which of the following statements are correct?
a. a E (-∞,1] [2,-) b.be (-0,0] [1, )
c. a= 1 + b
d. none of these
11
1233 ( mathbf{0}<mathbf{x}<mathbf{2} boldsymbol{pi} ; mathbf{0}<mathbf{y}<mathbf{2} boldsymbol{pi} ) and
( 3^{sin x+cos y}=1 ) and ( 25^{sin x^{2}+cos ^{2} y}=5 ) then
( (x, y) ) is This question has multiple correct options
( ^{mathrm{A}} cdotleft(frac{7 pi}{6}, frac{pi}{3}right. )
В. ( left(frac{7 pi}{6}, frac{5 pi}{3}right) )
c. ( left(frac{11 pi}{6}, frac{pi}{3}right) )
D. ( left(frac{11 pi}{6}, frac{5 pi}{3}right) )
11
1234 Illustration 6.21 Solve V5 –14 V10+245 = 8,76(0,5).
sin x
COS X
11
1235 65. If seco + tano = 5, then the value
tan 0+1
is
tan 0 – 1
11
1236 If ( tan theta=frac{1}{2} ) and ( tan phi=frac{1}{3}, ) then the
value of ( boldsymbol{theta}+boldsymbol{phi} ) is
A ( cdot frac{pi}{6} )
в. ( pi )
c. ( frac{pi}{4} )
D.
11
1237 ULL
17.
Let f(O)=sin(sino+sin30). Then f (0) is (2000)
(a) 20 only when 020 (6) < 0 for all real 0
© 20 for all real e (d) <0 only when 0 <0
11
1238 The sum of all the solutions of the
equation ( cos theta cos left(frac{pi}{3}+thetaright) cos left(frac{pi}{3}-thetaright)= )
( frac{1}{4}, theta epsilon[0,6 pi] )
A. ( 15 pi )
в. ( 30 pi )
c. ( frac{100 pi}{3} )
D. None of these
11
1239 Fin an acute angle ( Theta ), when
( frac{cos Theta-sin Theta}{cos Theta+sin Theta}=frac{1-sqrt{3}}{1+sqrt{3}} )
11
1240 13. Which of the following is not the general solution of
2cos2x + 1 = 3.2-sin?x?
a. nn, ne Z
b. n + – 1 ,ne Z
2)
it , ne z
d. none of these
2
11
1241 4. In A ABC, if sin’e = sin(A – ) sin(B – 0) sin(C – 0), then
prove that cot 0= cot A + cot B + cot C.
11
1242 From the following exact of the sine
table, the value of ( sin 37^{0} 27^{prime} ) is equal to
( 12^{prime} 18^{prime} 24^{prime} 30^{prime} 36^{prime} 42^{prime} 48^{prime} 54^{prime} 1^{prime} 2 )
A . 0.6075
B. 0.6081
c. 0.6088
D. 0.6115
11
1243 18. If 3 sin ß= sin (2a + B), then tan (a +B) – 2 tan a is
Tot a. independent of a
b. independent of ß
c. dependent of both a and
B a r
d. independent of both a and ß
11
1244 If ( cos A=frac{4}{5} ) find ( tan A ) 11
1245 The solution of the equation
( (sin x+cos x)^{1+sin 2 x}=2,-pi leq x leq pi )
is
A ( cdot frac{pi}{2} )
B.
c. ( frac{pi}{4} )
D. none of these
11
1246 47. If tan 30+ tan 0 = 2 tan 20, then 0 is equal to (ne zo
na
no a. nn OOTDT b.
d. none of these
11
1247 55.
If tan(a coso) = cot (nt sino), then
will be equal to (O se
11
1248 Find the degree measures corresponding to the following radian measures ( left(boldsymbol{U s e} boldsymbol{pi}=frac{mathbf{2 2}}{mathbf{7}}right) )
(i) ( frac{11}{16} )
(ii) -4
(iii) ( frac{5 pi}{3} )
(iv) ( frac{7 pi}{6} )
11
1249 43. If cos 25° + sin 25º = p, then cos 50° is
a. 12 – p? b. – √2-p²
c. p /2-p2. d.-pſ2-p²
rata
11
1250 If ( 0 leq x leq pi, ) the interval in which the
function ( f(x)=frac{tan x}{sin x} ) is defined is:
A. ( 0 leq x leq pi )
В. ( 0<x<pi )
c. ( 0<x<frac{pi}{2} )
D ( cdot frac{pi}{2}<x leq pi )
E ( cdot 0<x<frac{pi}{2} ) and ( frac{pi}{2}<x<pi )
11
1251 Find the value of ( frac{cos A-sin A+1}{cos A+sin A-1}- )
( (operatorname{cosec} A+cot A) )
11
1252 Assertion
The number of real solution of the
equation ( sin (cos x)=cos (sin x) ) is
zero
Reason
( sin x>0, ) then ( 2 n pi<x< )
( (2 n+1) pi, n epsilon I )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion false but Reason is true
11
1253 11. Let tanx-tan-x >0 and 2sin x nt,ne
Z b . x >nt – Tc/6, n e Z
c. x<nt – Te/4, ne Z d. x<nt + tc/6, ne z
11
1254 48. If cose sin € + V sin’e + sin’a} Sk, then the value of k
is
on
a. Vi+cos’
a
b
d
. V1+sinʼa
. √2 + cos²a
c. √2+ sin²a
11
1255 Evaluate: ( int_{0}^{frac{pi}{2}} sqrt{1+sin x} d x ) 11
1256 x +

7. The number of solutions of the equatio
CO.
+ cos x – 2 cos (x+”). cos* = sin? – in interval
11
1257 Find the value of ( cos theta-cos 3 theta+cos 4 theta )
when ( boldsymbol{theta}=mathbf{4 5}^{circ} )
A. ( frac{sqrt{3}}{2} )
B. ( sqrt{3}-1 )
( c cdot frac{1}{2} )
D. ( sqrt{2}-1 )
11
1258 The possible value of ( boldsymbol{theta} in(mathbf{0}, boldsymbol{pi}) ) such
( operatorname{that} sin (theta)+sin (4 theta)+sin (7 theta)=0 ) are
A ( cdot frac{2 pi}{9}, frac{pi}{4}, frac{4 pi}{9}, frac{pi}{2}, frac{3 pi}{4}, frac{8 pi}{9} )
В. ( frac{pi}{4}, frac{5 pi}{12}, frac{pi}{2}, frac{2 pi}{3}, frac{3 pi}{4}, frac{8 pi}{9} )
С ( cdot frac{2 pi}{9}, frac{pi}{4}, frac{pi}{2}, frac{2 pi}{3}, frac{3 pi}{4}, frac{35 pi}{36} )
D. ( frac{2 pi}{9}, frac{pi}{4}, frac{pi}{2}, frac{2 pi}{3}, frac{3 pi}{4}, frac{8 pi}{9} )
11
1259 ( tan theta=-2, theta epsilon(0, pi) ) then which of the
following is correct
This question has multiple correct options
A ( cdot sin theta=frac{-2}{sqrt{5}} )
B. ( sin theta=frac{2}{sqrt{5}} )
( ^{mathrm{C}} cdot cos theta=frac{-1}{sqrt{5}} )
D. ( cos theta=frac{1}{sqrt{5}} )
11
1260 Find the value of ( cos 37^{circ} 16^{prime} ) 11
1261 Find the centroid of Triangle whose
vertices are
( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}(mathbf{2},-mathbf{1}, mathbf{6}), boldsymbol{C}(mathbf{3}, mathbf{2},-mathbf{3}) )
A ( .(2,1,2) )
В. (2,-1,2)
c. (-2,-2,-2)
D. (1,2,1)
11
1262 90. If cos A + cos²B + cos²C = 1, then A ABC is
a. equilateral
b. isosceles
c. right angled
d. none of these
11
1263 Which of the following statements is/are correct for ( 0<theta<frac{pi}{2} ? )
This question has multiple correct options
A ( cdot(cos theta)^{1 / 2} leq cos frac{theta}{2} )
B. ( (cos theta)^{3 / 4} geq cos frac{3 theta}{4} )
c. ( cos frac{5 theta}{6} geq(cos theta)^{5 / 6} )
D. ( cos frac{7 theta}{8} geq(cos theta)^{7 / 8} )
11
1264 Solve ( sin (50+theta)-cos (40-theta)+ )
( tan 1 tan 10 )
( tan 20 tan 70 tan 80 tan 89=1 )
11
1265 If ( tan 6 theta=frac{p}{q}, ) find the value of ( frac{1}{2}(p operatorname{cosec} 2 theta-q sec 2 theta) ) in terms of ( p )
and ( boldsymbol{q} )
( mathbf{A} cdot 2 sqrt{p^{2}+q^{2}} )
B ( cdot sqrt{p^{2}+q^{2}} )
c. ( frac{sqrt{p^{2}+q^{2}}}{q} )
D. ( frac{sqrt{p^{2}+q^{2}}}{p} )
11
1266 If (x – a) cos 0 + y sin 0= (x – a) cos 0 + y sin o = a and
tan (8/2) – tan (0/2) = 2b, then
a. y2 = 2ax – (1 – 62) x2
b. tan = 6 + bx)
² = 26x- (1 – 2) x²
d. tan 2-1 (y – bx)
2 x
2.
X
11
1267 The value of ( int_{0}^{frac{pi}{2}} frac{cos 3 x+1}{2 cos x-1} d x ) is
( A cdot 2 )
B.
( c cdot frac{1}{2} )
D.
11
1268 3. Prove that 5 cos 0 +
8 +-
+ 3 lies between – 4
(IIT-JEE 1979)
and 10.
11
1269 2. The maximum value of the expression
-is (IIT-JEE 2010)
sin? 0 + 3 sin o cos 0 + 5 cos? O
11
1270 If
( boldsymbol{theta} ) and ( phi ) are angles in the 1 st quadrant such that ( tan theta=frac{1}{7} ) and ( sin phi=frac{1}{sqrt{10}} )
( mathbf{A} cdot theta+2 phi=90^{circ} )
B . ( theta+2 phi=60^{circ} )
c. ( theta+2 phi=30^{circ} )
D. ( theta+2 phi=45^{circ} )
11
1271 Prove that:
( tan x+tan left(frac{pi}{3}+xright)-tan left(frac{pi}{3}-xright)= )
( 3 tan 3 x )
11
1272 Find the value of ( tan ^{-1}left(tan frac{2 pi}{3}right) ) 11
1273 Find ( ^{prime} x^{prime} ) if ( sec ^{2} 2 x=1-tan 2 x ) 11
1274 Find :
( cos frac{pi}{65} cos frac{2 pi}{65} cos frac{4 pi}{65} cos frac{8 pi}{65} cos frac{16 pi}{65} )
A ( cdot frac{1}{64} )
в. ( frac{1}{32} )
( c cdot frac{1}{16} )
D. None of these
11
1275 20. In a AABC, if tan A : tan B : tan C = 3: 4: 5, then the
value of sin A sin B. sin C is equal to
b. 215
T5
c. 215
d.
375
11
1276 n a right angled ( triangle A B D, angle B= )
( mathbf{6 0}^{circ}, angle boldsymbol{A}=mathbf{3 0}^{circ} )
Then ( sin 30^{circ} ) is equal to
( A )
( 3 frac{1}{2} )
( c )
( D cdot 1 )
11
1277 17. The number of solutions of the equation 1 + cos x +
cos 2x + sin x + sin 2x + sin 3x = 0, which satisfy the
condition
sa is
11
1278 Illustration 3.29
Prove that cos 55° + cos 65° + cos 175° = 0.
11
1279 62. If OSO 90° and cos20-sin30
= cos 90°, then will be equal
to
(1) 16° (2) 18°
(3) 20°
(4) 22°
11
1280 The value of ( frac{sin ^{2} 53+cos ^{2} 53}{sec ^{2} 37-tan ^{2} 37} ) is
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot frac{1}{4} )
D.
11
1281 Prove that: ( sec ^{6} boldsymbol{A}-tan ^{6} boldsymbol{A}=mathbf{1}+ )
( mathbf{3} tan ^{2} boldsymbol{A}+mathbf{3} tan ^{mathbf{4}} boldsymbol{A} )
11
1282 ( 160^{circ} ) in radian measure is
A ( cdot frac{2 pi^{c}}{5} )
в. ( frac{3 pi^{c}}{5} )
c. ( frac{8 pi^{c}}{9} )
D. ( pi c )
11
1283 If ( sin (theta+alpha)=a ) and ( sin (theta+beta)= )
( boldsymbol{b},((mathbf{0}<boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{theta}<boldsymbol{pi} / 2)) ) then
( 2 cos ^{2}(alpha-beta)-1-4 a b cos (alpha-beta) ) is
A ( cdot 1-a^{2}-b^{2} )
B . ( 1-2 a^{2}-2 b^{2} )
c. ( 2+a^{2}+b^{2} )
D . ( 2=a^{2}-b^{2} )
11
1284 ootan
99. The value of a
norna
o
cos
in-
2
1
1
b.
b._2
sin 20
ē
. sin 200
C. sin 20
0
d. sin o
ő
11
1285 Illustration 3.57
If tan
olm
| 01
tan
ola
prove that
Vatb
cosa=
a cos o +b
a+bcos o
11
1286 Consider the geometric progression ( boldsymbol{S}=mathbf{1}+boldsymbol{2} sin ^{2} boldsymbol{theta}+boldsymbol{4} sin ^{4} boldsymbol{theta}+boldsymbol{8} sin ^{6} boldsymbol{theta}+ )
( ldots . ) up to infinite terms, where ( mathrm{S} ) is a finite number and ( boldsymbol{theta} neq frac{boldsymbol{n} boldsymbol{pi}}{mathbf{2}} ) where ( mathbf{n} varepsilon mathbf{I} )
Then Values of ( theta ) always lies in the
interval?
A ( cdotleft(-frac{pi}{6}, frac{pi}{6}right) )
в. ( left(0, frac{pi}{3}right) )
c. ( left(-frac{pi}{3}, 0right) )
D. ( left(-frac{pi}{4}, frac{pi}{4}right)-{0} )
11
1287 Value of the expression ( (1-cos Theta)(1+ )
( cos Theta)left(1+cot ^{2} Thetaright) ) is
A .
B.
( c cdot sin ^{2} theta )
D. ( operatorname{cosec}^{2} Theta )
11
1288 70. The value of sin2 65° + sin2 25°
+ cos2 35° + cos2 55° is
(1) O
(2) 1
(3) 2
11
1289 Illustration 4.52
Solve cos5°x – sin50x = 1.
11
1290 If ( a=frac{sin x times cos 3 x}{sin 3 x times cos x} ) then which of the
following is wrong?
A ( cdot a3 )
c. ( frac{1}{3}<a<3 )
D. None of these
11
1291 Express the ( 150^{0} ) in radians: 11
1292 7. If tan 60=plq, find the value of -(p cosec 20 – q sec 20)
in terms of p and q.
11
1293 Find the value of ( sin ^{2} 30^{circ}+cos ^{2} 60^{circ} ) 11
1294 The simplification of ( cos (A+ )
( B) cos (A-B) ) is equivalent to:
A ( cdot sin ^{2} A-sin ^{2} B )
B. ( cos ^{2} B-sin ^{2} A )
( mathbf{c} cdot cos ^{2} A-cos ^{2} B )
D. ( cos 2 A cdot sin 2 B )
11
1295 If ( tan alpha=2, ) then the value of
( frac{sin alpha}{sin ^{3} alpha+cos ^{3} alpha} ) is
( A cdot frac{2}{9} )
в. ( frac{5}{9} )
c. ( frac{10}{9} )
D. ( frac{5.5}{9} )
11
1296 10. If x + y = 27/3 and sin x/sin y = 2, then the
a. number of values of x € [0, 41] are 4
b. number of values of x € [0, 41] are 2
c. number of values of ye [0, 41] are 4
d. number of values of ye [0, 41] are 8
11
1297 n the given figure, ( angle boldsymbol{A}+angle boldsymbol{B}+angle boldsymbol{C}+ )
( angle D+angle E ) is equal to
2
( c cdot frac{3 pi}{pi} )
2
22
11
1298 28. The value of sin? 12° + sin? 21° + sin? 39° + sin? 48° –
sin? 9º – sin? 18° is
11
1299 Illustration 3.16
Prove that tan 70° = 2tan 50° + tan 20°.
11
1300 Illustration 2.16 By geometrical interpretation, prove that
i. sin(a+B) = sin a cos ß+ sin ß cos a
ii. cos(a + B) = cos a cos B- sin a sinß
11
1301 52. If 2 sin 20|=|tan B + cot Bl, a,ße 69,
value of a + Bis
then the
b. T
niel
d.
11
1302 Find the value of ( frac{cos 70^{0}}{sin 20^{0}}+ )
( cos 57^{0} operatorname{cosec} 33^{0}-2 cos 60^{0} )
11
1303 If ( alpha, beta, gamma, delta ) are in arithmetic
progression. Then which is of the following is correct
A. ( tan (alpha+delta)=tan (beta+gamma) )
B. ( tan (alpha+gamma)=tan (beta+delta) )
c. ( tan (alpha+beta)=tan (gamma+delta) )
D. none of these
11
1304 2. Number of roots of the equation sinx cos xl +
| 2 + tanx + cotx = 3 ,xe 0, 41, are
11
1305 4xy
15. sec 0 =
(x + y)2 15
2 is true if and only if (1996 – 1 Mark
(a) x+y=0
(b) x=y,x=0
(d) x 70, y0
(c) x=y
11
1306 The value of the expression ( left(1+cos frac{pi}{10}right)left(1+cos frac{3 pi}{10}right)left(1+cos frac{7 pi}{10}right)(1 )
is
11
1307 ( mathbf{A}=cos 20^{0} cos 40^{0} cos 60^{0} cos 80^{0} )
( mathbf{B}=cos 6^{0} cos 42^{0} cos 66^{0} cos 78^{0} )
( mathbf{C}=cos mathbf{3} mathbf{6}^{mathbf{0}} cos mathbf{7} mathbf{2}^{mathbf{0}} cos mathbf{1 0} mathbf{8}^{mathbf{0}} cos mathbf{1} mathbf{4} mathbf{4}^{mathbf{0}} )
A. ( A>B>C )
в. ( B>C>A )
c. ( C>A>B )
D. ( A=B=C )
11
1308 Solve the following equations. ( sin frac{x}{2} cos 2 x+sin ^{2} x cos frac{x}{2}= )
( cos ^{2} x cos frac{x}{2} )
11
1309 If ( theta ) is in the first quadrant and ( cos theta= )
( frac{3}{5}, ) then the value of ( frac{5 tan theta-4 operatorname{cosec} theta}{5 sec theta-4 cot theta} ) is
A ( cdot frac{5}{34} )
в. ( frac{5}{16} )
c. ( frac{5}{-34} )
D. ( frac{-5}{16} )
11
1310 If the tangents of the angles ( A ) and ( B ) of ( a ) triangle ( A B C ) satisfy the equation
( a b x^{2}-c^{2} x+a b=0, ) then
This question has multiple correct options
( mathbf{A} cdot tan A=frac{a}{b} )
B. ( tan B=frac{b}{a} )
( mathbf{c} cdot cos C=0 )
D. ( sin ^{2} A+sin ^{2} B+sin ^{2} C=2 )
11
1311 Solve the following equations. ( cos ^{6} x-sin ^{6} x=frac{13}{8} cos ^{2} 2 x ) 11
1312 Find the value of ( tan 25^{circ} 15^{prime} ) 11
1313 Find the distance between ( mathrm{P}( )
( a sin alpha,-b cos alpha) & Q(-a cos alpha, b sin alpha) )
11
1314 The number of solution of tan x+sec x=2cos x in [0,2 T) is
[2002]
(a) 2 (b) 3 C) 0 (d) 1
11
1315 Find the values of other trigonometric
function ( (sin , cos , tan ) ) in each of the
following problems
i) ( sin theta=3 / 5: theta ) in ( I^{s t} ) quadrant
ii) ( cos theta=-1 / 2: theta ) in ( I I^{s t} ) quadrant
iii) ( sin theta=3 / 4: theta ) in ( I I I^{s t} ) quadrant ( $ $ )
11
1316 ( A ) tower of ( x ) meters high has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant
( y ) meters from the foot of the tower then
the length of the flagstaff in meters is
A ( frac{yleft(x^{2}-y^{2}right)}{x^{2}+y^{2}} )
B. ( frac{xleft(y^{2}+x^{2}right)}{y^{2}-x^{2}} )
c. ( frac{xleft(x^{2}+y^{2}right)}{x^{2}-y^{2}} )
D. ( frac{xleft(x^{2}-y^{2}right)}{x^{2}+y^{2}} )
11
1317 If ( P=cos frac{pi}{20} cdot cos frac{3 pi}{20} cdot cos frac{7 pi}{20} cdot cos frac{9 pi}{20} & Q )
( =cos )
( frac{pi}{11} cdot cos frac{2 pi}{11} cdot cos frac{4 pi}{11} cdot cos frac{8 pi}{11} cdot cos frac{16 pi}{11}, ) ther
is
A. not defined
B. 1
( c cdot 2 )
D. none of these
11
1318 Illustration 3.75 If A + B + C = 180°, prove that cos-A +
cos-B + cos²C= 1-2 cos A cos B cos C.
COS
11
1319 If ( 0, alpha, beta<frac{pi}{4} ) such that ( cos (alpha+beta)=frac{4}{5} )
andsin ( (alpha-beta)=frac{5}{13}, ) then the value of
( tan 2 alpha= )
A. ( frac{56}{65} )
в. ( frac{56}{23} )
c. ( frac{56}{33} )
D. ( frac{56}{36} )
11
1320 1-
X
then value of
57. If tan 62° = 1+ x
tan 208° will be
(1)
1-x²
1+ 2
1 + x2
1-x²
1
X
+
x
11
1321 Illustration 4.38
Find the number of roots of the equation
tan x+3) = 2 tan x, for IE (0, 3m).
11
1322 ( frac{sin 30^{0}+tan 45^{0}+operatorname{cosec} 60^{0}}{sec 30^{0}+cos 60^{0}+cot 45^{0}}= ) 11
1323 Find the value of ( 2 sin 3 theta cos theta- )
( sin 4 theta-sin 2 theta )
11
1324 IF ( theta ) is in the first quadrant and ( cos theta= )
3. then value of
( overline{mathbf{5}} )
( 5 tan theta-4 operatorname{cosec} theta sec theta-4 cot theta ) is
A ( cdot frac{-14}{3} )
в. ( frac{5}{16} )
( c cdot frac{5}{-34} )
D. ( frac{-5}{16} )
11
1325 =1, then AABC is
+tan 2 TI-B
– + tan 2 T-C
13. If tan21-A
4 ou 4
a. equilateral
ollers c. scalene
b. isosceles
d. none of these 2011 SS
11
1326 ( ln operatorname{acircle} ) of radius ( 21 mathrm{cm}, ) an arc
subtends an angle of ( 60^{circ} ) at the
centre.Find
(i) the length of the arc
( (i i) ) the area of the sector
(iiii) the area of the minor segment and
( (i v) ) the area of the major segment.
11
1327 Illustration 3.74 Prove that 2 sin 2° + 4 sin 4° + 6 sin 6° +
… + 180 sin 180° = 90 cot 10°.
11
1328 ( (sin theta+csc theta)^{2}+(cos theta+sec theta)^{2} ) is
( A cdot geq 9 )
B . ( leq 9 )
( mathrm{c} cdot=9 )
D. None of these
11
1329 If ( boldsymbol{A}=mathbf{6 0}^{circ} ) and ( boldsymbol{B}=mathbf{3 0}^{boldsymbol{o}}, ) verify that
( tan (boldsymbol{A}+boldsymbol{B})=frac{tan boldsymbol{A}+tan boldsymbol{B}}{1-tan boldsymbol{A} tan boldsymbol{B}} )
11
1330 29. Value of 3+cot 80% cot 20°
cot 80°+cot 20°
o is equal to
a. cot 20°
b. tan 50°
c. cot 50°
d. cot 20°
11
1331 The value of ( x ) fo which ( sin (pi x)+ )
( cos (pi x)=0 )
11
1332 If ( cos theta+sin theta=sqrt{2}, ) find the value of
( cos theta-sin theta )
11
1333 48. The solution of 4 sin’x + tan-x + cosec?x + cotx – 6=0
is (n e 2)
a. nt +
b. 2nd
Bit Bim
c. Na +
d. nr –
11
1334 ( I f cos 10^{circ} cos 30^{circ} cos 50^{circ} cos 70^{circ}=x )
Find ( 16 x )
11
1335 The number of solution of the equation ( |cot x|=cot x+frac{1}{sin x} operatorname{in}[0,2 pi] ) is
( A cdot 2 )
B. 4
c. 0
( D )
11
1336 21. If tan x + tan 2x + tan 3x = tan x tan 2x tan 3x then value
of sin 3x + cos 3x| is
11
1337 If ( sin theta=0.47, ) then ( sin (pi-theta)= )
A . -0.47
B . -0.43
c.
D. 0.43
E . 0.47
11
1338 Illustration 3.47 Let f(x) = 2 cosec 2x + sec x + cosec x. Then
find the minimum value of f(x) for x el
11
1339 33. General solution of sinx – 5 sinx cos x – 6 cos²x = 0 is
a. x=nt – Te/4, n e Z only
b. nt + tan-‘ 6, ne Z only
c. both (a) and (b)
o consavia
d. none of these
11
1340 Solve the equation ( sin theta+sin 3 theta+sin 5 theta=0 ) 11
1341 4. If cot(a + B) = 0, then sin(a + 2B) can be
a. – sin a
b. sin ß
c. cos a
d. cos ß
11
1342 3. Sum of roots of the equation x4 – 2x² sin? * + 1 = 0 is
a. O
c. 1
b. 2
d. 3
11
1343 2. For which values of a does the equation 4 sin(x + Tt/3)
cos(x – /6) = 0? + V3 sin 2x – cos 2x have solutions?
Find the solutions for a = 0, if any exists.
11
1344 The measure of an angle in degrees,
grades and radians be ( mathrm{D}, mathrm{G} ) and ( mathrm{C} )
respectively, then relation between them ( frac{boldsymbol{D}}{mathbf{9 0}}=frac{boldsymbol{G}}{mathbf{1 0 0}}=frac{boldsymbol{2 C}}{boldsymbol{pi}} ) but ( mathbf{1}^{circ}= )
( left(frac{180}{pi}right)^{0} simeq 57^{circ}, 17^{prime}, 44.8^{prime prime} ) and sum of
interior angles of a ( n ) -sided regular polygon is ( (2 n-4) frac{pi}{2} . ) On the basis of above information, answer the following
questions :The angles between the hour hand and minute hand of a clock at half
past three is –
( A cdot frac{pi}{3} )
B. ( frac{pi}{4} )
c. ( frac{5 pi}{12} )
D. ( frac{7 pi}{12} )
11
1345 17.
Prove that the values of the function
ne
sin x cos 3x
sin 3 x cos x
between and 3 for any real:
100% Made
between
and 3 for any real x.
(1997 – 5 Marks)
11
1346 ( cos ^{3} theta+cos ^{3}left(120^{circ}+thetaright)+cos ^{3}left(120^{circ}-right. )
( boldsymbol{theta})= )
( A cdot frac{3}{4} sin 3 theta )
B . ( frac{3}{4} cos 3 theta )
( mathrm{c} cdot frac{3}{4} tan 3 theta )
D. ( frac{3}{4} cot 3 theta )
11
1347 11.
If A=sinx + cos4x, then for all real x:
[2011
(b) 1SAS2
(a) sasi
11
1348 1. If f(0) =
1-sin 20 + cos 20
-, then value of
2 cos 20
8f (11°) • f (34°) is
11
1349 The number of solutions of the equation
( sin x=cos 3 x ) in ( [0, pi] ) is
A . 1
B . 2
( c .3 )
D.
11
1350 44. The number of roots of (1 – tan 2) (1 + sin 20) = 1 + tan e
for 0 [0, 21] is
a. 3
d. none of these
b. 4
c. 5
11
1351 Find the angle made by a ladder of length ( 4 m ) with the ground if its one end is ( 2 m ) away from the wall and the other end is on the wall 11
1352 89.
sin 2 A+sin 2B + sin 2C
is equal to be
sin A+sin B + sin C
A B C abs
a. 8sin
sin – sin b. 8 cos -COS
Ola Ola
18
c. 8 tan
– tan
– tan
8 cot — cot –
11
1353 if ( tan 25^{circ}=x, ) then
( frac{tan 155^{circ}-tan 115^{circ}}{1+tan 155^{circ} tan 115^{circ}} ) is equal to
A ( cdot frac{1-x^{2}}{2 x} )
в. ( frac{1+x^{2}}{2 x} )
c. ( frac{1+x^{2}}{1-x^{2}} )
D. ( frac{1-x^{2}}{1+x^{2}} )
11
1354 31. The number of solutions of the equation sin’x cos x +
sin?x cos²x + sin x cos x = 1 in the interval [0, 271) is/are
b. 2
c. 3
end. infinite
a. 0
11
1355 sin a + sin B = and cos a + cos B =
7. The value of sin(a+B) is
d. none of these
11
1356 13. For x e(0, Tt), the equation sin x + 2 sin 2x – sin 3x = 3
has
ban S
sur
a. infinitely many solutions
b. three solutions
c. one solution
d. no solution
(JEE Advanced 2014)
11
1357 What is most general value of ( theta ) which
satisfies both the equations. ( sin theta=-1 / 2 ) and ( tan theta=1 / sqrt{3} )
11
1358 General solution of ( sin ^{3} x+cos ^{3} x+ )
( frac{3}{2} sin 2 x=1 )
A. ( x=n pi ) when ( n ) is even integer
B. ( x=2 n pi ) when ( n ) is odd integer
C ( cdot x=n pi+frac{pi}{2} ) when ( n ) is odd integer
D・ ( x=n pi-frac{pi}{2} )
11
1359 69. If O be acute and tan 0 + cot 0 =
2, then the value of tans 0 + cot10
O is
(1) 1
(2) 2
(3) 3
(4) 4
11
1360 Illustration 3.45 If sin A = 3/5 and 0° < A < 90°, find the
values of sin 2A, cos 2A, tan 2A, and sin 4A.
11
1361 16. If 0 < x < 21, then the number of real values of x, which
satisfy the equation cos x + cos2x+cos 3x + cos 4x =0 is:
[JEEM 2016
(2) 7
(b) 9
(c) 3
(d) 5
1.
fon Av
.
11
1362 Illustration 4.31 Solve the equation 2(cosx + cos 2x) + sin
2x(1+2 cosx) = 2 sinx for x (-10 < x < 0).
11
1363 If ( 2 sin theta+1=0 ) and ( sqrt{3} tan theta=1 ) then
find general value of ( boldsymbol{theta} ) is
( ^{mathrm{A}} cdot_{n pi pm} frac{pi}{6} )
в. ( quad n pi+(-1)^{n} cdot frac{7 pi}{6} )
c. ( _{2 n pi+frac{7 pi}{6}} )
D. ( 2 n pi+frac{11 pi}{6} )
11
1364 The general solution of ( tan x-sin x= )
( 1-tan x sin x )
A ( cdot x=n pi+frac{pi}{4} )
( x=n pi+(-1)^{n}left(-frac{pi}{2}right) )
В ( cdot x=frac{n pi}{4}-frac{pi}{4} )
( x=n pi+(-1)^{n}left(-frac{pi}{2}right) )
C ( cdot x=n pi+frac{pi}{4} )
D. ( x=n pi+frac{pi}{6} )
( x=n pi+(-1)^{n}left(-frac{pi}{2}right) )
11
1365 If ( mathbf{p}_{1}, mathbf{p}_{2}, mathbf{p}_{3} ) are the principal values of following trigonometric equations ( sin theta=-frac{1}{sqrt{2}} )
2) ( cos theta=-frac{sqrt{3}}{2} )
3) ( tan theta=sqrt{3}-2 )
A. ( p_{1}<p_{2}<p_{3} )
в. ( p_{1}<p_{3}<p_{2} )
c. ( p_{3}<p_{1}<p_{2} )
D . ( p_{2}<p_{3}<p_{1} )
11
1366 If tan 0 + sec 0 = 1.5, find sin e, tan , and
Illustration 2.6
sec e.
11
1367 Prove that
( frac{sin A+sin B}{cos A+cos B}=tan frac{A+B}{2} )
11
1368 Solve: ( |cos x|=cos x-2 sin x )
A ( cdot x=(2 n+1) pi+frac{pi}{4} )
B ( cdot x=(2 n+1) pi+frac{pi}{2} )
C ( cdot x=2 n pi+frac{pi}{4} )
D ( cdot x=2 n pi+frac{pi}{2} )
11
1369 If ( (2 cos x+sin x)=1, ) then sum of all
possible value of ( (7 cos x+6 sin x) )
is
11
1370 15. If sin x + sin y 2 cos a cos x ve R, then sin y + cos a is
equal to
11
1371 The general solution of the equation
( sin x+cos x=1 ) is
A ( cdot x=2 n pi+frac{pi}{2}, n=0,pm 1,pm 2 )
B ( cdot x=n pi+left((-1)^{n}+1right) frac{pi}{4}, n=0,pm 1,pm 2 )
C ( x=n pi+left((-1)^{n}-1right) frac{pi}{4}, n=0,pm 1,pm 2 )
D. ( x=2 n pi, n=0,pm 1,pm 2 )
11
1372 If ( A+B+C=pi ) then the expression ( frac{sin 2 A+sin 2 B-sin 2 C}{sin 2 A+sin 2 B+sin 2 C} ) reduces to 11
1373 11 UUD 100
8. Prove that tan a + 2 tan 20 + 4 tan 4a + 8 cot 8a=cot a.
(IIT-JEE 1988)
11
1374 Illustration 4.15 Find the number of solutions of [cos x] +
sin x = 1 in a <x<37 (where [.] denotes the greatest integer
function).
11
1375 The number of values of x in the interval [0, 511] satisfying
the equation 3 sinx-7 sin x +2=0 is (1998 – 2 Marks)
(a) 0 (b) 5
(c) 6
(d) 10 |
11
1376 sin 0 + cos 0
69. If
ue of sin”e – cose is
sin 0 –coso = 3, then the val-
11
1377 24. If cot (0 – 0), 3 coté, cot (0+ a) are in A.P. and is not
4 sine
an integral multiple of 5, then the value of 4S
3sinº a
11
1378 23. The general solution of the equation 8 cos x cos 2x cos 4x
= sin 6x/sin x is
a. x = (n/7) + (1/21), ne z
b. x = (21/7) + (10/14), Vnez
c. x = (n/7) + (1/14), Vnez
d. x = (n )+ (1/14), V ne z
11
1379 If ( frac{a x}{cos theta}+frac{b y}{sin theta}=a^{2}-b^{2} ) and
( frac{a x sin theta}{cos ^{2} theta}-frac{b y cos theta}{sin ^{2} theta}=0 ) then ( (a x)^{2 / 3}+ )
( (b y)^{2 / 3}=left(a^{2}-b^{2}right)^{2 / 3} ? )
A. True
B. False
11
1380 If ( sin a=frac{3}{7} ) and ( cos a<0, ) what is the
value of ( tan a )
A ( cdot frac{-(3 sqrt{20})}{10} )
B. ( frac{-(sqrt{10})}{10} )
c. ( frac{-(3 sqrt{10})}{20} )
D. ( frac{(2 sqrt{5})}{10} )
11
1381 Illustration 3.96
5x +12y+ 7xy
If x2 + 12 = x2,2 then find the range of
11
1382 The terminal arm is in II quadrant, what
are the measures of possible angles?
A . In between ( 90^{circ} ) and ( 180^{circ} ) or ( -270^{circ} ) and ( -180^{circ} )
B. In between ( 180^{circ} ) and ( 270^{circ} ) or ( -90^{circ} ) and ( -180^{circ} )
C. In between ( -90^{circ} ) and ( 0^{circ} ) or ( 270^{circ} ) and ( 360^{circ} )
D. None of these
11
1383 The value of ( sin 75^{circ}= )
( ^{A} cdot frac{2-sqrt{3}}{sqrt{2}} )
в. ( frac{sqrt{3}+1}{2 sqrt{2}} )
c. ( frac{sqrt{3}-1}{2 sqrt{2}} )
D. ( frac{sqrt{3}+1}{sqrt{2}} )
11
1384 Solve the equations
( 3left(sec ^{2} theta+tan ^{2} thetaright)=5 )
11
1385 If the angle ( alpha ) lies in the first quadrant
and ( tan alpha+cot alpha=2, ) then the value
of ( sqrt{tan alpha}+sqrt{cot alpha} ) is
A . -4
B. – –
( c cdot 2 )
D. 4
11
1386 3.
Let n be an odd integer. If sin no = Eb sin’
r=0
value of 0, then
(a) b=1, b, = 3
(c) b. =-1, b =n
an odd integer. If sin ne = { b sin” e, for every
(1998 – 2 Marks)
(b) b =0,b=n
(d) b = 0, b, = n2 – 3n+3
11
1387 ( tan 9-tan 27-tan 63+tan 81= ) 11
1388 66. The total number of solutions of log sin x| = -x + 2x in
[0, 1] is equal to
a. 1
b. 2
c. 4
d. none of these
11
1389 Solve:
( frac{tan theta}{sec theta+1}+frac{tan theta}{sec theta-1}=2 csc theta )
11
1390 63. If sino + coseco = 2, then val-
ue of sin 1000 + cosec1009 is
equal to :
(1) 1
(2) 2
(3) 3
(4) 100
11
1391 If ( 8 tan A=15, ) then the value of ( frac{sin A-cos A}{sin A+cos A} ) is:
A ( cdot frac{7}{23} )
в. ( frac{11}{23} )
c. ( frac{13}{23} )
D. ( frac{17}{23} )
11
1392 [
begin{aligned}
operatorname{Let} A &=left{theta: 2 cos ^{2} theta+sin theta leq 2right} text { and } \
B &={theta: pi / 2 leq theta leq 3 pi / 2} . text { Then }
end{aligned}
]
find the value of ( boldsymbol{A} cap boldsymbol{B} )
11
1393 9. If sin x + cosec x = 2, then sin”x + cosec”x is equal to
a. 2
b. 21
c. 2n-1
d. 2n-2
11
1394 ( (sin theta+cos theta)(1-sin theta cos theta) ) can be
written as:
( mathbf{A} cdot sin theta+cos theta )
( mathbf{B} cdot sin ^{3} theta-cos ^{3} theta )
( mathbf{c} cdot sin ^{3} theta+cos ^{3} theta )
( mathbf{D} cdot sin theta-cos theta )
11
1395 Assertion ( (A): sin ^{2} theta+sin ^{2}left(theta+60^{0}right)+ )
( sin ^{2}left(theta-60^{0}right)=frac{3}{2} )
Reason ( (mathrm{R}): cos alpha+cos left(120^{0}+alpharight)+ )
( cos left(120^{0}-alpharight)=0 )
A. Both A and R are true and R is the correct explanation to A
B. Both A & R are true but R is not the correct explanation to ( A )
c. A is true, R is false
D. A is false, R is true
11
1396 Sin 2B
85. If tan(a-B) =
• If tanſa-B) –
Cos 2R, then
a. tan a=2 tan B
c. 2 tan a=3 tan ß
b. tan B=2 tan a
d. 3 tan a= 2 tan ß
11
1397 360° 540°
56. cosec – + cosec
=
7
1800
90°
a. cosec
b. cosec
– 2001
180°
c. sec
900
d. sec
11
1398 If ( cos theta=-frac{3}{5}, pi^{c}<theta<frac{3 pi^{c}}{2}, ) find the
value of ( frac{csc theta+cot theta}{sec theta-tan theta} )
11
1399 If ( cos A=frac{4}{5}, cos B=frac{12}{13}, frac{3 pi}{2}< )
( A, B<2 pi, ) find the values of the
following.
( (mathrm{i}) cos (boldsymbol{A}+boldsymbol{B}) )
(ii) ( sin (A-B) )
11
1400 6. Find the smallest positive root of the equation
sin(1 – x) = cos x .
11
1401 If ( boldsymbol{theta}+boldsymbol{phi}=frac{boldsymbol{pi}}{boldsymbol{4}}, ) then ( (mathbf{1}+tan boldsymbol{theta})(mathbf{1}+ )
( tan phi) ) is equal to
A .
B. 2
( c cdot frac{5}{2} )
D.
11
1402 Solve: ( 4 cos ^{2} x+6 sin ^{2} x=5 )
A ( cdot x=n pi+frac{pi}{2} )
B. ( x=n pi-frac{pi}{3} )
c. ( x=n pi pm frac{pi}{4} )
D. None of these
11
1403 What is the value of ( sin ^{2} 25^{0}+sin ^{2} 65^{0} ? ) 11
1404 The equation ( sec ^{2} theta=frac{4 x y}{(x+y)^{2}} ) is only
possible, when
( mathbf{A} cdot x=y )
в. ( xy )
D. None of these
11
1405 If both the distinct roots of the equation
( |sin x|^{2}+|sin x|+b=0 ) in ( [0, pi] ) are
real, then the value of ( b ) is
( mathbf{A} cdot[-2,0] )
B . (-2,0)
( c cdot[-2,0) )
D. None of these
11
1406 Simplify, using trigonometric tables
( tan 63^{circ} 12^{prime}-cos 12^{circ} 42^{prime} )
11
1407 42. The value of k if the equation 2 cos x + cos 2kx = 3 has
only one solution is
b. 2
d. 1/2
a. O
c. 12
11
1408 If ( cos (A-B)=frac{3}{5} & tan A tan B=2 )
then
This question has multiple correct options
A ( cdot cos A cos B=frac{1}{5} )
B. ( sin A sin B=-frac{2}{5} )
( c cdot cos (A+B)=-frac{1}{5} )
D. ( sin A sin B=frac{2}{5} )
11
1409 1
43. Number of solution(s) satisfying the equation
sin x
sin 4x in [0, 47] equals
sin 2x
a. 0
c. 4
b. 2
d. 6
11
1410 Prove that:
[
begin{array}{c}
frac{tan theta-cot theta}{sin theta cos theta}=sec ^{2} theta-operatorname{cosec}^{2} theta= \
tan ^{2} theta-cot ^{2} theta
end{array}
]
11
1411 Prove that ( frac{sin theta-cos theta+1}{sin theta+cos theta-1}= )
( frac{1}{sec theta-tan theta}, ) using the identity
( sec ^{2} theta=1+tan ^{2} theta )
11
1412 Show that ( tan ^{2} theta-frac{1}{cos ^{2} theta}=-1 ) 11
1413 Prove that ( : sqrt{left(sec ^{2} Theta+operatorname{cosec}^{2}right.}= )
( tan Theta+cot Theta )
11
1414 Express in radians ( 345^{circ} 25^{prime} 36^{prime prime} )
A . ( 1.726268 pi^{c} )
B . ( 1.3465338 pi^{c} )
( mathbf{c} cdot 1.91903 pi^{c} )
D. ( 1.1978258 pi^{c} )
11
1415 3.
Which one is not periodic
(a) sin3x|+sinºx
(c) cos 4x + tan²x
[2002]
(b) cos Vx + cos x
(d) cos2x + sinx
11
1416 In a circle diameter, ( 40 mathrm{cm} ), the length of a chord is ( 20 mathrm{cm} . ) Find the length of
minor arc of the chord.
11
1417 Evaluate
( 8 sqrt{3} operatorname{cosec}^{2} 30^{circ} sin 60^{circ} cos 60^{circ} cos ^{2} 45^{circ} )
A ( cdot 2 sqrt{3} )
B. ( 4 sqrt{3} )
( c cdot 8 sqrt{3} )
D. ( 12 sqrt{3} )
11
1418 f ( operatorname{cosec} A=2, ) find the value of
( frac{1}{tan A}+frac{sin A}{1+cos A} )
11
1419 5. Number of solutions of the equation
(√3+ 1)² + (13-1 * = 23r is
11
1420 9.
Find the values of xe(-1, + 7) which satisfy the equation
8(1+cos xl+ cos2xl+cosº xlt…) – 43 (1984 – 2 Marks)
11
1421 Which one of the following is correct?
( mathbf{A} cdot sin 45^{0} cos 45^{0}=1 )
B ( cdot sin ^{2} 45^{0}-cos ^{2} 45^{0}=1 )
( mathbf{C} cdot sin 30^{0}+cos 60^{0}=1 )
D ( cdot cos ^{2} 30^{0}-cos ^{2} 60^{0}=1 )
11
1422 83. If both the distinct roots of the equation (sin xß+ |sin x! +
b = 0 in [0, 1] are real, then the values of b are
a. [-2, 0]
b. (-2,0)
c. [-2,0)
d. none of these
11
1423 Find the domain of definition of the
following function:
( y=sqrt{sin ^{2} x-sin x} )
11
1424 If ( boldsymbol{m} sin boldsymbol{theta}=boldsymbol{n} sin (boldsymbol{theta}+boldsymbol{2} boldsymbol{alpha}), ) then
( tan (boldsymbol{theta}+boldsymbol{alpha}) cdot cot boldsymbol{alpha} ) equal to
A. ( frac{1-n}{1+n} )
в. ( frac{m+n}{m-n} )
c. ( frac{m-n}{m+n} )
D. ( frac{1+n}{1-n} )
11
1425 3. Which of the following quantities are rational? 11
1426 Illustration 4.10 If 2tan” x – 5sec x = 1 for exactly seven
distinct values of x € [0, nm/2], n e N then find the greatest
value of n.
11
1427 78. In triangle ABC, if angle C is 90° and the area of triangle
is 30 sq. units, then the minimum possible value of the
hypotenuse c is equal to
a. 3072
b. 6012
c. 12072
d. 2730
11
1428 If an angle ( a ) is divided into two parts ( A )
& ( B ) such
that ( A-B=x ) and ( tan A: tan B= )
( K: 1, ) then the value of ( sin x ) is
A ( cdot frac{k+1}{k-1} sin alpha )
B. ( frac{k}{k+1} sin alpha )
c. ( frac{k-1}{k+1} sin alpha )
D. none of the above
11
1429 Solve:
( frac{cos 70}{sin 20}+cos 59 csc 31 )
11
1430 7. Let 0<,<0,<0z< … denote the positive solution of the
equation 3 + 3 cos 0 = 2 sin? . The value of 03 + 0, is
a. 61
b. 71
c. 87
d. 411
11
1431 64. Ifre ( 7, 37), then 4 cos ( 7
) + VasinⓇx+ sin? 2x
is always equal to
a. 1
C.-
2
y equal cos
b. 2
d. none of these
Pone of these
020
11
1432 3. Find the values of x € (-, 1) which satisfy the equation
8 (1+cos xl+cos xl+cos” x + …) = 43
(IIT-JEE 1984)
11
1433 Assertion
The system of linear equations ( boldsymbol{x}+(sin boldsymbol{alpha}) boldsymbol{y}+(cos boldsymbol{alpha}) boldsymbol{z}=mathbf{0} )
( x+(cos alpha) y+(sin alpha) z=0 quad ) has a non
( -boldsymbol{x}+(sin boldsymbol{alpha}) boldsymbol{y}-(cos boldsymbol{alpha}) boldsymbol{z}=mathbf{0} )
trivial solution for only one value of ( boldsymbol{alpha} )
Iying between 0 and ( pi )
Reason ( left|begin{array}{ccc}sin x & cos x & cos x \ cos x & sin x & cos x \ cos x & cos x & sin xend{array}right|=0 ) has no
solution in the interval ( -frac{pi}{4}<x<frac{pi}{4} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
1434 Find principal and general solution of the equation, ( cot x=-sqrt{3} ) 11
1435 The number of distinct solutions of
( sin 5 theta cdot cos 3 theta=sin 9 theta cdot cos 7 theta ) in ( theta in )
( [mathbf{0}, boldsymbol{pi} / mathbf{2}] ) is
( A cdot 4 )
B. 5
( c cdot 8 )
D.
11
1436 75. Given that a, b, c are the sides of a AABC which is right
angled at C, then the minimum value of -+-
a. 0
c. 6
b. 4
d. 8
200
11
1437 Prove that ( frac{cos (pi+x) cos (-x)}{sin (pi-x) cos left[frac{pi}{2}+xright]}= )
( cot ^{2} x )
11
1438 55. The total number of solutions of cosx = V1-sin 2x in
[0, 21) is equal to
b. 3
c. 5
d. none of these
a. 2
11
1439 ( frac{1-sin A}{cos A} ) is equal to
( A cdot frac{cos A}{1+sin A} )
B. ( frac{sin A}{1-cos A} )
c. ( frac{tan A}{1+tan A} )
D. ( frac{tan A}{1+cos A} )
11
1440 Illustration 4.25
Solve cos 0+ cos 30 – 2 cos 20= 0.
11
1441 A minimum value of ( sin x cos 2 x ) is-
( A cdot 1 )
B. – 1
c. ( -2 / 3 sqrt{6} )
D. None of these
11
1442 Illustration 3.86 Prove that
cos 20° cos 40° cos 60° cos 80º = 1/16.
11
1443 The total number of solutions of
( cos x=sqrt{1-sin 2 x} ) in ( [0,2 pi] ) is equal
to
A . 2
B. 3
c. 5
D. None of these
11
1444 Find the number of solutions of the
equations;
( mathbf{2}^{cos x}=|sin x| ) when ( boldsymbol{x} epsilon[-2 pi, 2 pi] )
11
1445 81. The equation cos*x + b cos x + 1 = 0 will have a solution
if b belongs to
a. (-,2]
b. [2,00)
c. (-, -2]
d. none of these
11
1446 tv
11. If Ois eliminated from the equations x = a cos(-a) and
ad y = b cos(0-3), then
me to say
(a-B) is equal to
a b² ab
a. sec (a – B) b. cosec°(a – b)
c. cos?( – B) d. sin?(a-B)
COS (a – B
11
1447 Find the general solution of ( x cos ^{2} 2 x+ )
( cos ^{2} 3 x=1 )
A ( cdot(2 k+1) frac{pi}{10}, k in I )
В ( cdot(pi+1) frac{pi}{10} ; k in I )
( c cdot(2 k-1) frac{pi}{10}, k in I )
D. Both (A) and (C)
11
1448 Illustration 4.33
Solve 2 tan 0-cot =-1.
11
1449 46. If 3 tan(0-159) = tan(@+ 15º), then 0 is equal to (ne z
a. nit +
b. nt +
+
c. nt +
d. none of these
11
1450 Which one of the following is the value of ( cos 170^{0} cos 10^{0}-sin 170^{0} sin 10^{0} ? )
A . -2
B. – 1
( c . )
D.
11
1451 If ( 2 sin 2 theta=sqrt{3}, ) then ( tan theta ) is
( A )
B. ( frac{1}{sqrt{3}} )
( c cdot sqrt{3} )
D. ( frac{1}{sqrt{2}} )
11
1452 Find the area of the isosceles triangle
with base ( 16 mathrm{cm} ) and vertical angle
( mathbf{6 0}^{circ} mathbf{4 0}^{prime} )
11
1453 16. If (sin a) x2 – 2x + b 2 for all the real values of x = 1
and a e (0, T/2) U (F/2, T), then the possible real values
of b is/are
a. 2
b. 3
c. 4
d. 5
11
1454 31. Range of f(0) = cos? O (cos2 0 + 1) + 2 sin? O is
a. [3/4, 1]
b. [3/16, 1]
c. [3/4, 7/4]
d. [7/4, 2]
11
1455 Find the ratio of ( sin x, cos x, tan x, ) where
( mathbf{x}=mathbf{4 5}^{mathbf{0}} ? )
11
1456 ( A B C ) is a triangle in which ( A B= )
( A C=4 mathrm{cm} ) and ( angle A=90^{circ} . ) Calculate
the length of perpendicular from ( boldsymbol{A} ) to ( boldsymbol{B C} )
A. ( 2.83 mathrm{cm} )
B. ( 1.414 mathrm{cm} )
( c .2 .6 mathrm{cm} )
D. 2.20 cm
11
1457 Show that ( frac{sin 2 alpha+sin 2 beta}{cos 2 alpha-cos beta}=cot (beta- )
( boldsymbol{alpha} )
11
1458 Illustration 2.59
Show that tan 1° tan 2° … tan 89° = 1.
11
1459 If ( cos ^{2} x+cos ^{2} 2 x+cos ^{2} 3 x=1 ) then
A ( cdot x=(2 n+1) frac{pi}{4}, n epsilon I )
B. ( x=(4 n+1) frac{pi}{4}, n in I )
c. ( x=n pi frac{pi}{4}, n epsilon I )
D. none of these
11
1460 The incorrect statement is
( mathbf{A} cdot sin theta=-frac{1}{5} )
( mathbf{B} cdot cos theta=1 )
( mathbf{c} cdot sec theta=frac{1}{2} )
( mathbf{D} cdot tan theta=20 )
11
1461 2 TV
Illustration 3.71
Find the value of cos
6T
+cos — +cos —
11
1462 Illustration 4.54
Solve 1 + sinx sin? – =0.
11
1463 Given that ( cos 50^{circ} 18^{prime}= )
0.6388 and ( cos 50^{circ} 42^{prime}=0.6334, ) then
the possible value of ( cos 50^{circ} 20^{prime} ) is
A . 0.6293
B. 0.6307
c. 0.636
D. 0.6414
11
1464 Prove that: ( cos ^{2} boldsymbol{alpha}+cos ^{2}(boldsymbol{alpha}+boldsymbol{beta})- )
( boldsymbol{2} cos boldsymbol{alpha} cos boldsymbol{beta} cos (boldsymbol{alpha}+boldsymbol{beta})=sin ^{2} boldsymbol{beta} )
11
1465 ( 110^{circ} 30^{prime} ) in radians is :
A ( cdot frac{221 pi^{c}}{360} )
в. ( frac{225 pi^{c}}{360} )
c. ( frac{231 pi^{c}}{360} )
D. ( frac{229 pi^{c}}{360} )
11
1466 Equation ( 6 sin ^{2} theta-5 sin theta+1=0 ) is
satisfied by
A ( cdot theta=frac{pi}{2} )
B. ( theta=frac{pi}{3} )
( mathbf{c} cdot theta=frac{pi}{4} )
D・ ( theta=frac{pi}{6} )
11
1467 Find which of the number of the form
( left(n pi-t a n^{-1} 3right), ) where ( n epsilon l, ) are
solution for ( 12 tan 2 x+frac{sqrt{10}}{cos x}+1=0 )
( mathbf{A} cdot n=k, k epsilon z )
B. ( n=(3 k+1), k epsilon z )
c. ( n=(2 k+1), k epsilon z )
D. ( n=(4 k+1), k epsilon z )
11
1468 Illustration 2.60
Find the value of cos? *+ cos20
16
16
2777
+ cos-
CO2 In
16
+ cos2
16
11
1469 17. Find the number of solutions of 8 € [0, 21] satisfying the
equation (log/z tan e) (Vlogan o 3 + log/5 33) — 1.
11
1470 54. The equation sinfx + cos x + sin 2x + a=0 is solvable for
a. – 5/2 sas 1/2 b. – 3 sasi
c. – 3/2 s as 1/2 d. – 1 sasi
11
1471 ( cos x+cos y=frac{4}{5}, cos x-cos y=frac{2}{7} )
The value of ( 14 tan left(frac{x-y}{2}right)+ )
( mathbf{5} cot left(frac{boldsymbol{x}+boldsymbol{y}}{mathbf{2}}right) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot frac{5}{4} )
D.
11
1472 Which of the following is correct?
( A cdot sin 1^{circ}>sin 1 )
B. ( sin 1^{circ}<sin 1 )
( mathbf{c} cdot sin 1^{circ}=sin 1 )
D. ( sin 1^{circ}=frac{pi}{180} sin 1 )
11
1473 Suppose the point with coordinates (-12,5) is on the terminal side of angle
( theta, ) the value of the sine trigonometric function of ( theta . ) is ( frac{a}{13} ) Find ( a )
11
1474 Solve ( frac{sin 30^{circ}+tan 45^{circ}-cos e s 60^{circ}}{sec 30^{circ}+cos 60^{circ}+cos 45^{circ}} ) 11
1475 Illustration 3.85 The product of the sines of the angles of a
triangle is p and the product of their cosines is q. Show that
the tangents of the angles are the roots of the equation qx –
px + (1 + q) x-p= 0.
11
1476 The value of the expression 13 cosec 20°-sec 20° is
equal to
(1988-2 Marks)
(a) 2
(b) 2 sin 20%sin 40°
(c) 4
(d) 4 sin 20%sin 40°
11
1477 If ( tan theta=frac{x sin phi}{1-x cos phi}, tan phi= )
( frac{y sin theta}{1-y cos theta}, ) then ( frac{x}{y}= )
A. ( frac{sin phi}{sin theta} )
B. ( sin theta )
sin ( phi )
c. ( frac{1-cos phi}{1-cos theta} )
D. ( frac{1-cos theta}{1-cos phi} )
11
1478 Solve ( tan theta=2, ) then ( theta=n pi+ )
( a, ) where ( a=tan ^{-1}(2), n in I )
If true then enter 1 and if false then
enter 0
11
1479 Solve
( 7 sin ^{2} theta+3 cos ^{2} theta=4 )
11
1480 TT
37
3. The value of (1 + com 5 ) (+ cos?(1 + cos *)
3. The value of
+ cos-
+ cos
+ COS
210
a. 1/4
c. 1/8
b. 3/4
d. 3/8
(IIT-JEE 1984)
11
1481 Find the value of other five
trigonometric ratios:
( sec x=frac{13}{5}, x ) lies in fourth quadrant
11
1482 If ( sec theta=1 ; 0 leq theta<12^{circ}, ) then the value
of ( boldsymbol{theta} ) is
A ( .5^{circ} )
B . 0
( c cdot 1^{c} )
D. ( 2^{circ} )
11
1483 ii.

Illustration 3.40 Prove that
sin20
sin 20
= tan
– = coto
1 + cos2e
1- cos 20
1 + sin 20 + cos20
1 + sino – cos o
== tan 0/2
1+ sin 20 – cos 20
1+ sin 0 + cose
cos 20
– = tan (T/4 – ) vị.
= tan
1 + sin 20
1 + sine
= cot Oiv.
cos e
A12
11
1484 [2012]
24. The equation esinx – e-sinx_4=0 has :
(a) infinite number of real roots
(b) no real roots
© exactly one real root
(d) exactly four real roots
11
1485 If ( operatorname{cosec} x-c o t x=frac{1}{3}, ) where ( x neq 0 )
then the value of ( cos ^{2} x-sin ^{2} x ) is
A ( cdot frac{16}{25} )
в. ( frac{9}{25} )
( c cdot frac{8}{25} )
D. ( frac{7}{25} )
11
1486 If ( alpha, beta ) are two different values of ( theta ) lying
between 0 and ( 2 pi ) which satisfy the
equation ( 6 cos theta+8 sin theta=9 . ) Find
( sin (alpha+beta) )
11
1487 If ( A=tan 6^{0} tan 42^{0} ) and ( B= )
( cot 66^{0} cot 78^{0}, ) then
( A cdot A=2 B )
B. A=1/3 B
( c cdot A=B )
D. 3A=2B
11
1488 If ( tan alpha=2 sqrt{2}, ) then the value of
( frac{tan alpha}{frac{sin ^{3} alpha}{cos alpha}+sin alpha cdot cos alpha} )
( mathbf{A} cdot mathbf{0} )
B. 2
c. ( 2 sqrt{2} )
( D )
11
1489 If ( cos B cos C+sin B sin C sin ^{2} A=1 )
then triangle ( A B C ) is
A. isosceles and right angled
B. equilateral
C. isosceles whose equal angles are greater than ( pi / 4 )
D. none
11
1490 The value of ( left(1+tan ^{2} thetaright) sin ^{2} theta ) is
( mathbf{A} cdot sin ^{2} theta )
B. ( cos ^{2} theta )
( mathbf{c} cdot tan ^{2} theta )
( D cdot cot ^{2} theta )
11
1491 ( sqrt{3} sin theta=cos theta, ) find the value of
( frac{3 cos theta+2 sin theta}{2 cos ^{2} theta-2} )
11
1492 18. 1-r cos a
1-r cos O is equal to
r sin a
a. tan 20
c. sin 20
b. cot 20
d. cos 20
11
1493 5. For 0 <$<
, the solution(s) of
È cosec (0+ (m = 1, cosec( 6 + mut) = 472
4
m=1
is (are)
a. 7/4
c. 7/12
b. 7/6
d. 57/12
(IIT-JEE 2009)
11
1494 In ( Delta A B C, 4 sin A cos B=1 ) and
( tan B=3 tan A . ) Then ( sin C= )
A ( cdot frac{1}{4} )
в. ( frac{1}{2} )
( c cdot frac{3}{4} )
D.
11
1495 If A lies in the second quadrant and ( mathbf{3} tan boldsymbol{A}+mathbf{4}=mathbf{0}, ) the value of ( mathbf{2} cot boldsymbol{A}- )
( 5 cos A+sin A ) is equal to
A ( cdot-frac{53}{10} )
в. ( frac{23}{10} )
( c cdot-frac{37}{10} )
D. ( frac{7}{10} )
11
1496 77. The value of tan 6° tan 42° tan 66° tan 78° is
a. 1
b. 1/2
c. 1/4
d. 1/8
11
1497 Illustration 2.43 For real values of e, which of the following
is/are always positive?
a. cos(cos )
b. cos(sin )
:
c. sin(cos )
d. sin(sin )
11
1498 In A ABC, prove that cos A + cosB +
Illustration 3.100
cos C < 3/2.
11
1499 A wheel makes 360 revolutions in one
minute. Through how many radians does it turn in one second?
11
1500 Show that
( (sec theta-tan theta)^{2}=frac{1-sin theta}{1+sin theta} )
11
1501 Illustration 2.46
Find the range of f(x) = cos²x + sec?x.
11
1502 72. The value of tan 75° is:
13+1 13-1
(1) T3 – 1
(2) 73+1
1-13 1+13
(3) 1+13 (4) 1-13
11
1503 12. Let o, o e [0, 21t] be such that 2 cose (1 – sin o) = sin e
(tang+cot)cose1, tan (272-e) >0 and
-1 <sine <-
, then o cannot satisfy
(2012)
m) o<«<
<<
11
1504 If ( tan theta_{1}=k cot theta_{2}, ) then ( frac{cos left(theta_{1}+theta_{2}right)}{cos left(theta_{1}-theta_{2}right)}= )
A. ( frac{1+k}{1-k} )
B ( cdot frac{1-k}{1+k} )
c. ( frac{k+1}{k-1} )
D. ( frac{k-1}{k+1} )
11
1505 tan(In 6) tan(In 2) tan(In 3)
23. If –
tan(In 6) – tan(In 2) – tan(In 3)
= k, then the value of k
is
11
1506 60. If cos x = tan y, cos y = tan z, cos z = tan x, then the value
of sinx is
a. 2 cos 18°
b. cos 18°
c. sin 18°
d. 2 sin 18°
11
1507 If ( boldsymbol{x}+boldsymbol{y}=frac{boldsymbol{pi}}{2}, ) then prove that ( cos (boldsymbol{x}+ )
( boldsymbol{y})=mathbf{0} )
11
1508 ( cos 60^{0} times cos 30^{0}-sin 60^{0} times sin 30^{0}=0 ) 11

Hope you will like above questions on trigonometric functions and follow us on social network to get more knowledge with us. If you have any question or answer on above trigonometric functions questions, comments us in comment box.

Stay in touch. Ask Questions.
Lean on us for help, strategies and expertise.

Leave a Reply

Your email address will not be published. Required fields are marked *