Find the values of each of the following:

Solution:

(i) Let $\sin ^{-1} \frac{1}{2}=y$

Then,

$\sin y=\frac{1}{2}$

$\therefore \tan ^{-1}\left\{2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right\}=\tan ^{-1}\{2 \cos 2 y\}$

$=\tan ^{-1}\left(2\left(1-2 \sin ^{2} y\right)\right)$

$=\tan ^{-1}\left\{2\left(1-2 \times \frac{1}{4}\right)\right\}$

$=\tan ^{-1}\left\{2 \times \frac{1}{2}\right\}$

$=\tan ^{-1} 1$

$=\frac{\pi}{4}$

$\therefore \tan ^{-1}\left\{2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right\}=\frac{\pi}{4}$

(ii)

We have

$\cos \left(\sec ^{-1} x+\operatorname{cosec}^{-1} x\right)$

$=\cos \frac{\pi}{2} \quad\left[\because \sec ^{-1} x+\operatorname{cosec}^{-1} x=\frac{\pi}{2}\right]$

$=0$

$\therefore \cos \left(\sec ^{-1} x+\operatorname{cosec}^{-1} x\right)=0,|x| \geq 1$