Trigonometrical Functions and Identities
Choose the most appropriate option (a, b, c or d).
Q 1.
If tan  = a 
(a) 2a,
Q 2.
1
then sec  - tan  is equal to
4a
1
2a
(b) 
2
sin2  
(b) x = y, x  0
cosec  
(b) |x| = |y|  0
(b) 28
(d) none of these
(c) 24
(d) none of these
If x = rsin .cos , y = rsin  . sin  and z = rcos  then the value of x 2 + y2 + z2 is independent of
(b) r, 
(c) r, 
(d) r
Let p = a cos  - b sin . Then for all real 
(a) p  a2  b2
Q 8.
(c) x + y = 0, x  0
If sin  + cosec  = 2 then the value of sin8 + cosec8 is equal to
(a) , 
Q 7.
(d) none of these
x2  y2
, where x  R, y  R, gives real  if and only if
x2  y2
(a) 2
Q 6.
1
,2a
2a
(d) x  0, y  0
(c) x = y
(c) |x| = |y|  0
(b) x = y
(a) x = y  0
Q 5.
(d)
(x  y)2
, where x  R, gives real  if and only if
4xy
(a) x + y = 0
Q 4.
(c) 2a
4xy
, where x  R, y  R, is true if and only if
(x  y)2
sec  =
(a) x + y  0
Q 3.
1
,2a
2a
(b) p   a2  b2
(c)  a2  b2  p  a2  b2 (d) none of these
If 0 <  < 180 then
2  2  2  .....  2(1  cos ) ,
there being n number of 2’s, is equal to
(a) 2cos
Q 9.

2n
The value of tan
(b) 2cos

2n 1


 2 tan + 4 is equal to
16
8
(c) 2cos

2n 1
(d) none of these
(a) cot
Q 10.
(b) cot
1
2
(b) 
The value of
(b) 5
9
8
(b) 0
The value of cosec 10 -
1
2
(b) 4  10
If cos4  . sec2 ,
8
(a) AP
(c) 2.
sin 20o
sin 40o
(d) 4.
sin 20o
sin 40o
(c) 4
(d) none of these
(c) -2
(d) none of these
(c) 4
(d) 8
(c) 0
(d) none of these
1
and sin4  .cosec2  are in AP then
2
6
1
8
6
and sin  . cosec  are in
2
(b) GP
(c) HP
(d) none of these


5
7
If tan , x and tan
are in AP and tan , y and tan
are also in AP then
9
9
18
18
(a) 2x = y
Q 18.
(d) none of these
The least value of cos2 - 6 sin . cos  + 3 sin2  + 2 is
cos  . cos ,
Q 17.
(c) -1
3 sec10o is equal to
(b) 2
(a) 4  10
Q 16.
(d) none of these
The minimum value of cos 2 + cos  of real values of  is
(a)
Q 15.

4
16




The maximum value of 1  sin      2cos     for real values of  is
4
4




(a) 
Q 14.
1
2
(b) 4
(a) 3
Q 13.
(c) cot
3 cosec 20 - sec 20 is equal to
(a) 2
Q 12.

16
The value of sin 78 - sin 66 - sin 42 + sin 6 is
(a)
Q 11.

8
(b) x > y
(c) x = y
If cos(x – y), cos x and cos(x + y) are in HP then cos x.sec
(a) 1
(b) 2
(c)
2
(d) none of these
y
equals
2
(d) none of these
Q 19.
If 2 sin . cos  . sin  = sin  . sin( + ) then tan , tan  and tan  are in
(a) AP
Q 20.
(b) GP
Let f() =
(b) tan 2
(b) 
If tan
(a) 
Q 23.
627
725
Q 26.
a
b
If cos  
If
(c)
1
2
(d) none of these
627
725
(c) -1
(b) 
x y

y x
b
a
(d) none of these
 
is equal to
2
a2  b2
(c)
(d) none of these

3
4
, cos ( + ) =
and ( – ) = then sin 2 is equal to
5
5
4
(b) 0
(c) 2
(d) none of these
1
1
1
1
x   ,cos    y   then cos( – ) is equal to
2 
x
2
y
(b) xy 
1
xy
(c)
1 x y 
  
2 y x
(d) none of these
2sin 
1  sin   cos 
is equal to
 then
1  sin   cos 
1  sin 
(a)
Q 27.
(b)
If 0 <   <
(a)
(d) none of these


and tan are the roots of the equation 8x 2 – 26x + 15 = 0 then cos( + ) is equal to
2
2
(a) 1
Q 25.
1
2
If sin  + sin  = cos  –  = b then tan
(a) 
Q 24.
(c) cos 2
5
cot 
and    
. Then the value of f() . f() is
1  cot 
4
(a) 2
Q 22.
(d) none of these
If tan  = a , where a is a rational number which is not a perfect square, then which of the
following is a rational number ?
(a) sin 2
Q 21.
(c) HP
1

(b) 
If |tan A| < 1, and |A| is acute then
(a) tan A
(b) –tan 3
(c) 1 – 
1  sin 2A  1  sin 2A
1  sin 2A  1  sin 2A
(c) cot A
(d) 1 + 
is equal to
(d) –cot A
Q 28.




tan .tan     .tan     is equal to
3

3

(a) tan 2
Q 29.
The set of all possible values of  in [–, ] such that
 
(a) 0, 
 2
Q 30.
(b) 3
(c)
(b) a2 + d2 = b2 + c2
1
3
(d) none of these
(c) a2 + b2 = c2 + d2
(d) ab = cd
(b) p 2  p2
(c) p  2  p2
(d) none of these
(b) 5
(c) 1
(d) none of these
(c)  5
(b) –1
(d)




If 0  , ,x   cos2n ,y   sin2n  andz   cos2n  then
2
n 0
n 0
n 0
(a) xyz = xz + y (b) xyz = xy + z (c) xyz = x + y + z
n
Q 38.
(d) None of these
If cos 2x + 2cos x = 1 then sin2 x(2 – cos2x) is equal to
(a) 1
Q 37.
A
B
A B
. Then a – b is equal to
sin cos
2
2
2
(c)  1
(b) 0
(d) none of these
If 3sin  + 4cos  = 5 then the value of 4sin  – 3cos  is
(a) 0
Q 36.
(c) –cot 3
If cos 20 – sin 20 = p then cos 40 is equal to
(a) p 2  p2
Q 35.
(b) tan 
If asec  – ctan  = d and bsec  + dtan  = c then
(a) a2 + c2 = b2 + d2
Q 34.
   
(d)  , 
 2 2




If tan   tan      tan      k 3 then k is equal to
3
3


(a) 1
Q 33.
1  sin 
is equal to sec  – tan  is
1  sin 
(c) [ –, 0]
Let a = cos A + cos B – cos (A + B) and b = sin
(a) 1
Q 32.
   
(b) 0,    ,  
 2 2 
(d) none of these
For all real values of , cot  – 2cot 2 is equal to
(a) tan 2
Q 31.
(c) tan3 
(b) tan 3
Let n be an odd integer. If sin n =
n
b b
r 0
r
r 0
r
(d) xyz = yz + x
sinr  for all real  then
5
Q 39.
(a) b0 = 1, b1 = 3
(b) b0 = 0, b1 = n
(c) b0 = – 1, b1 = n
(d) b0 = 0, b1 = n2 – 3n – 3
If cos 5 = acos  + bcos  then c is equal to
5
3
(a) – 5
Q 40.
(b) 1
If sin3x. sin 3x 
n
c
m0
(a) 4
Q 41.
The value of sin
(b)
The numerical value of sin
1
64
(d) none of these
(c)
1
4
(d)

7
is equal to
.sin
18
18
1
8
1
2
5 1
. 10  2 5
2
(b)
5 1
. 10  2 5
5 1
. 10  2 5
4
(c)
(d) none of these
5 5
2
(b)
5 5
4
(c)
1
5 5
2
(d) none of these



 3 sec
 4cos is
10
10
10
(b)
5
(c) 1
(d) none of these
(b) 0
(c) tan 50
(d) none of these
If , ,  and  be four angles of a cyclic quadrilateral then the value of cos  + cos  + cos  +
cos  is
(a) 1
Q 48.
(c)
The value of tan 20 + 2tan 50 – tan 70 is
(a) 1
Q 47.
2
The value of 2tan
(a) 0
Q 46.
(b)
1
16
The value of cos 9 – sin 9 is
(a) 
Q 45.
(d) none of these
The value of tan 63 – cot 63) is equal to
(a)
Q 44.
(c) 9

3
5
7
9
11
13
is equal to
.sn .sin .sin .sin .sin
.sin
14
14
14
14
14
4
14
(a) 1
Q 43.
(d) none of these
.cosmx is an identity in x, where cm’s are constant then the value of n is
(b) 6
(a) 1
Q 42.
m
(c) 5
(b) 0
(c) – 1
If 4n =  then cot .cot 2.cot 3… cot (2n  1) is equal to
(d) none of these
Q 49.
1
64
(b)
The value of cos
2
r 1
(a)
Q 52.
n
2
(b)
The value of sin
(c)
1
2
(d) none of these
(c)
n
1
2
(d) none of these
(c)
(b) 
n
2
(d) none of these
(c) –
(d) none of these
The number of real solution of the equation cos7 x + sin4 x = 1 in the interval [–, ] is
(b) 3
(c) 5
(d) none of these



If the solutions for  from the equation sin2  – 2sin  +  = 0 lie in   2n  ,2n  1   then the
6
6


set of possible values of  is
 5

(c)   ,  
 4

(b) (–, 1)
(d) [1]
If ABCD is a convex quadrilateral such that 4sec A + 5 = 0 then the quadratic equation whose
roots are tan A and cosec A is
(a) 12x2 – 29x + 15 = 0
2
(c) 12x + 11x – 15 = 0
Q 57.
1
28
The sum of the real roots of cos6x + sin4 x = 1 in the interval –  < x ≤  is equal to
5 
(a)  ,1
4 
Q 56.
n 1
2
(b) 0
(a) 2
Q 55.
(d)

3
5
 sin
 sin
 ... to n terms is equal to
n
n
n
(a) 0
Q 54.
1
16
r
is equal to
n
(a) 1
Q 53.
(c)
(b) 1
n 1
 cos
1
32

3
5
7
9
is
 cos
 cos
 cos
 cos
11
11
11
11
11
(a) 0
Q 51.
(d) none of these
The value of cos 12. cos 24. cos 36. cos 48. cos 72. cos 84 is
(a)
Q 50.
(c) 
(b) – 1
(a) 1
(b) 12x2 – 11x – 15 = 0
(d) none of these
If ABCD is a cyclic quadrilateral such that 12tan A – 5 = 0 and 5cos B + 3 = 0 then the quadratic
equation whose roots are cos C, tan D is
2
(a) 39x – 16x – 48 = 0
2
(b) 39x + 88x + 48 = 0
2
(c) 39x – 88x + 48 = 0
Q 58.
The number of real solution of the equation sin(e ) = 2 + 2x is
x
(a) 1
Q 59.
(b) 0
(c) 1
(b) px2 – x + p = 0
(b) p2 = q(q + 2)
(b) 5
(d) none of these
(c) p2 + q2 = 2q (d) none of these
(c) 6
(d) 10
If x = ,  satisfies both the equations cos2 x + acos x + b = 0 and sin2x + psin x + q = 0 then
relation between a, b, p and q is
(b) a2 + b2 = p2 + q2
(d) none of these
If 0 ≤ a ≤ 3, 0 < b ≤ 3 and the equation x2 + 4 + 3 cos (ax + b) = 2x heat at lest one solution then
the value if a + b is
(a) 0
(b)
The equation cos   x 
(a) p 
Q 67.
(c) px2 – 2x + p = 0


and cot is
2
2
The number of values of x in the interval [0, 5] satisfying the equation 3sin2x – 7sin x + 2 = 0 is
(c) 2(b + q) = a2 + p2 – 2
Q 66.
(d) infinite
If sec  and cosec  are the roots of x2  px + q = 0 then
(a) 1 + b + a2 = p2  q  1
Q 65.
(d) [0, ]
If sin  = p, where |p| ≤ 1 then the quadratic equation whose roots are tan
(a) 0
Q 64.
  
(c)   , 
 2 2
(b) [–, 0]
(b) 2
(a) p2 = q(q – 2)
Q 63.
(d) infinite
If esin x  e  sin x  4  0 then the number of real values of x is
(a) px2 + 2x + p = 0
Q 62.
(c) 2
2
(a) 0
Q 61.
x
The equation (cos p1)x + (cos p)x + sin p = 0 in x has redal roots. Then the set of values of p is
(a) [0, 2]
Q 60.
(d) none of these
1
2
If f(x) 

2
(c) 
(d) none of these
p
for all x  R has a real solution for . Then
x
(b) p 
1
4
(c) p 
1
4
(d) none of these
sin3x
, where x  n, then the rage of values of f(x) for real values of x is
sin x
(a) [–1, 3]
(b) (–, 1)
(c) (3, +)
(d) [–1, 3)
Q 68.
The set of values of k  R such that the equation cos 2 + cos  + k = 0 admits of a solution for 
is
 9
(a) 0, 
 8
Q 69.
(b) [0, +]
(d) none of these
The set of values of   R such that tan  + sec  =  holds for some  is
2
(a) (, 1)
Q 70.
(c) [–2, 0]
(c) 
(b) (–, –1]
(d) [–1, +)
If tan A + tan B + tan C = tan A . tan B. tan C then
(a) A, B, C must be angles of a triangle
(b) the sum of any two of A, B, C is equal to the third
(c) A + B + C must be an integral multiple of 
(d) none of these
Choose the correct options. One or more options may be correct.
Q 71.
If x = sin ( – ). sin ( – ), y = sin( – ). Sin ( – ) and
z = sin( – ). Sin ( – ) then
(a) x + y + z = 0 (b) x + y – z = 0 (c) y + z – x = 0 (d) x3 + y3 + z3 = 3xyz
Q 72.
sin
(a)
Q 73.
15
7
3
is equal to
.sin .sin
32
16
8
1
15
8 2 cos
32
1
If a 
(b) cos 15
1
4 2
cos ec

16
(d)
1
8 2
cos ec

32
(c) sin 15
(d) sin 15. cos 75
1
then for all ral x
5cos x  12sin x
(a) the least positive value of a is
(a) a ≤
Q 75.
(c)

8 sin
32
Which of the following is a rational numbers?
(a) sin 15
Q 74.
(b)
1
13
1
13
(b) the greatest negative value of a is –
(d) –
1
1
≤a≤
13
13
Let y = sin2x + cos4x. Then for all real x
(a) the maximum value of y is 2
(b) the minimum value of y is
3
4
1
13
(c) y ≤
Q 76.
1
4
(d) y  – 1
Let y = sin x. sin (60 + x). sin (60 – x). sin(60 – x). Then for al real x
(a) the minimum value of y is –
(c) y ≤
Q 77.
(b) then maximum value of y is 1
1
4
(d) y  – 1
Let f n() = tan

.(1 + sec )(1 + sec 2)(1 + sec 4)… (1 + sec 2n). Then
2
  
(b) f3    1
 32 
  
(a) f2    1
 16 
Q 78.
1
4
Let 0 ≤  <
2
  
(c) f4    1
 64 
  
(d) f5 
 1
 128 

and x = Xcos  + Ysin, y = Xsin – Ycos  such that
2
2
2
2
x + 4xy + y = aX + bY , where a,b are constants. Then
(b)  
(a) a = – 1, b = 3
Q 79.
If 7cos x – 4sin x = cos(x + ) <
If A + B =
24
25
(d)  = cos–1
7
25
(d) |cos A – cos B| =
1
(c)  = – 25
2
1
(c) (cos A  B)  
3
3
2 3
If tan  = a  0, tan 2 = b  0 and tan  + tan 2 = tan 3 then
cos3x. sin 2x =
(b) ab = 1
(a) a3 
(c) a + b = 0
(d) b = 2a
n
a
m 1
Q 83.

3

, be true for all x  R then
2
1
(b) | cos A  cosB |
3
(a) a = b
Q 82.
(d)  =

and cos A + cos B = 1 then
3
(a) cos(A  B) 
Q 81.
(c) a = 3, b = – 1
(b)  = sin–1
(a)  = 25
Q 80.

4
m
3
,a2  0
8
sin mx is an identity in x. Then
(b) n  6,a1 
1
2
(c) n  5,a1 
1
4
(d)  am 
3
4
If 1 + cos(x – y) = 0 then
(a) cos x – cos y = 0
(b) cos x + cos y = 0
(c) sin x + sin y = 0
(d) cos x + sin y = 1
Q 84.
If A  0, B > 0, A + B

and y = tan A. tan B then
3
(a) the maximum value of y is 3
(c) the maximum value of y is
Q 85.
cis
(b) the minimum value of y is
1
3
1
3
(d) the minimum value of y is 0
2 4
6
is equal to

 cos
7
7
7
(a) an integer
(b) a positive rational number
(c) a negative rational number
(d) an irrational number
Answers
1a
2b
3c
4d
5a
6a
7c
8a
9b
10b
11b
12c
13a
14c
15b
16a
17a
18c
19c
20c
21c
22a
23b
24a
25c
26b
27c
28b
29d
30b
31a
32b
33c
34b
35a
36a
37b
38b
39c
40b
41c
42v
43a
44c
45a
46b
47b
48a
49a
50c
51c
52b
53a
54b
55a
56b
57a
58b
59d
60a
61c
62b
63c
64d
65c
66b
67d
68a
69d
70c
71a,d 72a,d 73c
74a,b 75b,c 76a,c 77a,b,c,d
81c
83b,c 84c,d 85c
82a,c,d
78b,c 79a,b,d
80b,c