Question:
Find the value of $\tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]$
Solution:
Let $\sin ^{-1} \frac{1}{2}=x$. Then, $\sin x=\frac{1}{2}=\sin \left(\frac{\pi}{6}\right)$.
$\therefore \sin ^{-1} \frac{1}{2}=\frac{\pi}{6}$
$\therefore \tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]=\tan ^{-1}\left[2 \cos \left(2 \times \frac{\pi}{6}\right)\right]$
$=\tan ^{-1}\left[2 \cos \frac{\pi}{3}\right]=\tan ^{-1}\left[2 \times \frac{1}{2}\right]$
$=\tan ^{-1} 1=\frac{\pi}{4}$
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