The value of $2{\sec ^{ - 1}}2 + {\sin ^{ - 1}}\left( {\frac{1}{2}} \right)$ is

We have, $2{\sec ^{ - 1}}2 + {\sin ^{ - 1}}\frac{1}{2} = 2{\sec ^{ - 1}}\sec \frac{\pi }{3} + {\sin ^{ - 1}}\sin \frac{\pi }{6}$

$= 2 \cdot \frac{\pi }{3} + \frac{\pi }{6}$ [and $\left. {{{\sin }^{ - 1}}(\sin x) = x} \right]$

$= \frac{{4\pi + \pi }}{6} = \frac{{5\pi }}{6}$