If $\cos \left( {{{\sin }^{ - 1}}\frac{2}{5} + {{\cos }^{ - 1}}x} \right) = 0$, then $x$ is equal to

We have, $\cos \left( {{{\sin }^{ - 1}}\frac{2}{5} + {{\cos }^{ - 1}}x} \right) = 0$
$\Rightarrow$ ${\sin ^{ - 1}}\frac{2}{5} + {\cos ^{ - 1}}x = {\cos ^{ - 1}}0$

$\Rightarrow$ ${\sin ^{ - 1}}\frac{2}{5} + {\cos ^{ - 1}}x = {\cos ^{ - 1}}\cos \frac{\pi }{2}$

$\Rightarrow$ ${\sin ^{ - 1}}\frac{2}{5} + {\cos ^{ - 1}}x = \frac{\pi }{2}$

$\Rightarrow$ ${\cos ^{ - 1}}x = \frac{\pi }{2} - {\sin ^{ - 1}}\frac{2}{5}$

$\Rightarrow$ ${\cos ^{ - 1}}x = {\cos ^{ - 1}}\frac{2}{5}$
$therefore,$ $x = \frac{2}{5}$