Evaluate $\displaystyle : \tan ^{-1}\left[2 \sin \left(2 \cos ^{-1} \frac{\sqrt{3}}{2}\right)\right]$

\[\begin{aligned} & \tan ^{-1}\left[2 \sin \left(2 \cos ^{-1} \frac{\sqrt{3}}{2}\right)\right] \\ & =\tan ^{-1}\left[2 \sin \left(2 \times \frac{\pi}{6}\right)\right]=\tan ^{-1}\left[2 \sin \frac{\pi}{3}\right] \\ & =\tan ^{-1}\left[2 \times \frac{\sqrt{3}}{2}\right]=\tan ^{-1} \sqrt{3}=\frac{\pi}{3} \end{aligned} \]