## Inverse Trigonometric Functions Questions and Answers Part-4

1. $\sin^{-1}6x+\sin^{-1}6\sqrt{3}x=-\pi/2$
if x is equal to
a) -1/12
b) 1/6
c) 1/12
d) -1/6

Explanation: The given equation can be written as

2. If $A=\tan^{-1}\left(1/7\right)$    and $B=\tan^{-1}\left(1/3\right),$
then
a) $\cos2A=24/25$
b) $\cos2B=4/5$
c) $\cos2A=\sin 4B$
d) All of the Above

Explanation:

3. $\cos^{-1}x$   is equal to
a) $2\sin^{-1}\sqrt{\frac{1-x}{2}}$
b) $2\cos^{-1}\sqrt{\frac{1-x}{2}}$
c) $2\cos^{-1}\sqrt{\frac{1+x}{2}}$
d) Both a and c

Explanation: Let cos–1 x = y, so that x = cos y. Then

4. If $\sin\left[2\cos^{-1}\left\{\cot\left(2\tan^{-1}x\right)\right\}\right]=0,x>0$
then
a) x = 1
b) $x=\sqrt{2}+1$
c) $x=\sqrt{2}-1$
d) All of the Above

Explanation: The given equation can be written as

5. Number of solutions of the equation $\left(\tan^{-1}x\right)^{2}+\left(\cot^{-1}x\right)^{2}=\frac{5\pi^{2}}{8}$
is
a) 0
b) 1
c) 2
d) 3

Explanation:

6. If $\sin^{-1}\frac{2a}{1+a^{2}}+\sin^{-1}\frac{2b}{1+b^{2}}=2\tan^{-1}x$
then x=
a) $\frac{a-b}{1+ab}$
b) $\frac{a+b}{1-ab}$
c) $\frac{a-b}{1-ab}$
d) $\frac{a+b}{1+ab}$

Explanation:

7. If $y=2\tan^{-1}x+\sin^{-1}\frac{2x}{1+x^{2}}$      then
a) $-\pi/2< y<\pi/2$
b) $-3\pi/2< y< 3\pi/2$
c) $-\pi< y<\pi$
d) $-\pi/4< y< \pi/4$

Explanation:

8. If $0<\sin^{-1}x< 1$    and $1+\sin^{-1}x+\left(\sin^{-1}x\right)^{2}+....$       upto infinity =2 then x is equal to
a) $\pi/6$
b) $\pi/4$
c) $\pi/3$
d) none of these

Explanation:

9. If $\cos^{-1}\frac{p}{a}+\cos^{-1}\frac{q}{b}=\alpha$
then $\frac{p^{2}}{a^{2}}-\frac{2pq}{ab}\cos\alpha+\frac{q^{2}}{b^{2}}$
is equal to
a) $\sin^{2}\alpha$
b) $\cos^{2}\alpha$
c) $\tan^{2}\alpha$
d) $\cot^{2}\alpha$

10. The inequality $\sin^{-1}x>\cos^{-1}x$    holds for
b) $x\epsilon\left(0,1/\sqrt{2}\right)$
c) $x\epsilon\left(1/\sqrt{2},1\right)$