Mark the tick against the correct answer in the following:
$\tan ^{-1}\left\{2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right\}=?$
A. $\frac{\pi}{3}$
B. $\frac{\pi}{4}$
C. $\frac{3 \pi}{4}$
D. $\frac{2 \pi}{3}$
To Find: The value of $\tan ^{-1}\left\{2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right\}$
Let, $x=\tan ^{-1}\left\{2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right\}$
$\Rightarrow \mathrm{x}=\tan ^{-1}\left\{2 \cos \left(2\left(\frac{\pi}{6}\right)\right)\right\}\left(\because \sin \left(\frac{\pi}{6}\right)=\frac{1}{2}\right)$
$\Rightarrow \mathrm{x}=\tan ^{-1}\left(2 \cos \frac{\pi}{3}\right)$
$\Rightarrow \mathrm{x}=\tan ^{-1}\left(2\left(\frac{1}{2}\right)\right)=\tan ^{-1} 1=\frac{\pi}{4}\left(\because \cos \left(\frac{\pi}{3}\right)=\frac{1}{2}\right.$ and $\left.\tan \left(\frac{\pi}{4}\right)=1\right)$