Inverse Trigonometric Functions — Class 12 Maths Solution

We have
${\tan ^{ - 1}}\left\{ {2\cos \left( {2{{\sin }^{ - 1}}\left( {\frac{1}{2}} \right)} \right)} \right\} = {\tan ^{ - 1}}\left\{ {2\cos \left( {2 \times \frac{\pi }{6}} \right)} \right\}$

$= {\tan ^{ - 1}}\left\{ {2\cos \frac{\pi }{3}} \right\} = {\tan ^{ - 1}}\left[ {2 \times \frac{1}{2}} \right] = {\tan ^{ - 1}}1 = \frac{\pi }{4}$