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Q. The value of $\tan ^{-1}\left[2 \sin \left(2 \cos ^{-1} \frac{\sqrt{3}}{2}\right)\right]$ is

Inverse Trigonometric Functions

Solution:

Given, $ \tan ^{-1} {\left[2 \sin \left(2 \cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)\right)\right] }$
$ =\tan ^{-1}\left[2 \sin \left(2 \times \frac{\pi}{6}\right)\right]$
$ =\tan ^{-1}\left[2 \sin \left(\frac{\pi}{3}\right)\right] $
$ =\tan ^{-1}\left[2 \times \frac{\sqrt{3}}{2}\right]$
$ =\tan ^{-1} \sqrt{3}=\frac{\pi}{3}$