EXERCISE-A

1. Prove the following :

120 5 3 16 1 7
(i) tan 1  2 sin 1 (ii) 2 cos1  cot 1  cos1  
119 13 13 63 2 25

41  1 x 1 x
(iii) cot 1 9  cos ec 1  (iv) cos1 x  2 sin 1  2 cos1
4 4 2 2

1 1 2
1 1 2 n1
2. Find the sum of the series tan  tan ............. tan .......... 
3 9 1  2 2 n1
1 b c 
3. In a ABC if A  900 , then prove that tan  tan 1  .
ca ab 4
2 
4. Solve for x and y : sin-1x + sin-1 y = and cos-1 x - cos-1 y =
3 3

5. Find the solution set of the equation, 3 cos-1 x = sin

1
e1  x c4x  1hj.
2 2

1 1  3x 2
1
6. Solve : the 3 tan x  cos  0.
(1  x 2 )3 / 2

3x  x 3
1 1
7. Find the interval for which 3 tan x  sin is independent of x
(1  x 2 )3/ 2

 1 1  2 (a  x) (x  b)  ab
 sin   bx
 2  ab 2
ax  
8. Prove that : cos1 =  .
ab   1 1  2 (a  x) (x  b)  a  b
 2  2 sin  a  b

2
xa
  

F
G1 xI
J F1  x I 2

and   arc sinG Jfor 0 < x < 1, prove that      . What the
9. If   2arc tan
H1  x K H1  x K 2

value of    will be if x > 1.

1 3
10. Find all possible value of p and q for which cos p  cos 1 1  p  cos 1 1  q  .
4
k2 -1 -1 2
11. Find the integral values of k for which the system of equations ; cos x + (sin y) = and
4

-1 -1
4 2
cos x . (sin y) = and possesses solutions and find those solutions.
16
12. Prove the followings :

–1
2t
–1 –1
3t  t 3  1 1 
(i) tan t + tan = tan for t  (-, -1)    ,   (1, )
1 t2 1  3t 2  3 3
2t 3t  t 3 1 
(ii) –1
tan t + tan –1
= - + tan–1
for t    1,  
1 t2 1  3t 2  3
2t 3t  t 3 1 
(iii) tan–1 t + tan–1 =  + tan–1
for t   , 1
1 t2 1  3t 2  3 
 3 sin 2  1   
13. Evaluate : tan-1   + tan-1  tan   for   0, 
 5  3 cos 2  4   2

xy  {1  x 2 ).(1  y 2 )}
14. Find the condition for which equation sin 1 x  cos 1 y  tan 1 is true.
y (1  x 2 )  x 1  y 2

1 2( x  y) (1  xy )
15. Find the condition for which equation tan 1 x  tan 1 y  sin 1 is true.
2 (1  x 2 ) (1  y 2 )
 EXERCISE-B

(WRITE-UP)

I Read the passage and answer the questions from 4 to 8.

Using given definition answer the following
We know that corresponding to every bijection (one-one onto function) f : A B there exists a
bijection g : B A defined by
g(y) = x if and only if f(x) = y
The function g : B A is called the inverse of function f : A B and is denoted by f–1
Thus, we have
f(x) = y f –1 (y) = x
We have also learnt that
(f –1 of) (x) = f –1 (f(x)) = f –1(y) = x, for all x A
(fof –1) (y) = f(f –1(y)) = f(x) = y, for all y B.
We know that trigonometric function are periodic functions and hence, in general, all trigonometric
functions are not bijectives. Consequently, their inverse do not exist. However, if we restric their
domains and co-domains, they can be made bijections and we can obtain their inverse.
The Domain and Range of the functions are given below :
Function Domain Range
–1
(i) sin x [–1, 1] [  / 2,  / 2]
–1
(ii) cos x [–1, 1] [0,  ]
–1
(iii) tan x R (–  /2,  /2)
–1
(iv) cosec x ( ,  1]  [1,  ) [–  /2,  /2] – {0}
(v) sec–1 x ( ,  1]  [1,  ) [0,  ] – {  /2}
(vi) cot–1 x R (0,  )

1. The value of sec–1 (sec ) = , for all belonging to

(A) [0, ] – {/2} (B) (0, ) – {/2} (C) (0, ) (D) none of these

2. The value of tan–1 (tan ) = – 2, for all belonging to

(A) R (B)  (C) (3/2, 5/2) (D) [/2, 3/2]

3. The value of x, if 2 sin–1 x = sin–1 (2x 1 x 2 ), is

 1   1 1 
(A) [–1, 1] (B)   ,1 (C)   , (D) none of these
 2   2 2 

4. The value of sin–1 (sin 12) + cos–1 (cos 12)is

(A) 0 (B) /2 (C) –/2 (D) none of these

5. The value of sin–1 [cos (cos–1 (cos x) + sin–1 (sin x))], where x (/2, ) is
(A) /2 (B) –/2 (C)  (D) –
 EXERCISE-C

(MATCHING)

1. List - I List-II

 1  1  3 5
A. sin   sin     1.
3  2  2

 1
 3   24
B. cos  cos   2   6  2.
    7

1 1 5
C. tan  cos  3. 1
2 3 

 1 4
D. tan  2sin 4. –1
 5 
A B C D A B C D
(a) 1 3 4 2 (b) 4 3 4 2
(c) 3 4 1 2 (d) 3 2 1 4
 PROBLEMS

OBJECTIVE

 1  4  1  2  
1. The value of tan cos    tan    is
 5  3 
6 7
(A) (B)
17 16
16
(C) (D) none of these
7

 2 
2. The principal value of sin–1  sin  is
 3 
2 2
(A)  (B)
3 3
4
(C) (D) none of these
3

3. If we consider only the principle values of the inverse trigonometric functions then the value of
 1 1 4 
tan  cos  sin 1  is
 5 2 17 

29 29
(A) (B)
3 3

3 3
(C) (D)
29 29


4. The number of real solutions of tan-1 x(x  1)  sin 1 x 2  x  1  is
2
(A) zero (B) one
(C) two (D) infinite

 x 2 x3   2 x 4 x6  
5. –1
If sin  x    ...  + cos–1  x    ....  = for 0 < |x| <
2 4 2 4 2 , then x equals
    2
(A) 1/2 (B) 1
(C) –1/2 (D) –1

6. The value of x for which sin (cot–1 (1 + x)) = cos(tan–1x) is

(A) 1/2 (B) 1
(C) 0 (D) –1/2
 SUBJECTIVE
1
1. Find the value of : cos (2cos–1 x + sin–1x) at x = , where 0  cos–1 x   and –/2  sin–1 x  /
5
2.

-1 -1
x2 1
2. Prove that cos tan sin cot x = .
x2  2

1  2x  5x 2   
3. Find the range of values of t for which 2 sin t = 2 , t  2 , 2  .
3x  2x  1  
 ANSWERS

EXERCISE-A

L
M 3 O
N2 P
 1
Q 6. x   
2. 4. x = , y= 1 5. ,1 3, 0
4 2

 1   1  1
7. x    ,

   ,
3  3 
9. – 10. 0  p  1 and q 
2

4 2
11. K = 2 ; cos , 1 and cos ,  1 14.  1  x  y  1or  1  x  y  1
4 4
xy
15. xy<1 and  1  1
1  xy
EXERCISE-B

1. (A) 2. (C) 3. (C) 4. (A) 5. (B)

EXERCISE-C

1. (C)
PROBLEMS

OBJECTIVE

1. (D) 2. (D) 3. (D) 4. (C) 5. (B)

6. (D)

SUBJECTIVE
2 6
1. 
5
     3  
3.   2 , 10    10 , 2 
   