$\sin \left\{ {{\sin }^{ - 1}}\frac{1}{2} + {{\cos }^{ - 1}}\frac{1}{2} \right\} = $

Statement-$1$: ${\cot ^{ - 1}}\left[ {\frac{{\log (e/{x^2})}}{{\log (ex^2)}}} \right] + {\cot ^{ - 1}}\left[ {\frac{{\log (ex^2)}}{{\log (e/{x^2})}}} \right] = \frac{\pi}{2}$
Statement-$2$: ${\tan ^{ - 1}}\left[ {\frac{{1 + \log {x^2}}}{{1 - \log {x^2}}}} \right] = {\tan ^{ - 1}}1 + {\tan ^{ - 1}}(\log {x^2})$

If $\sin^{-1} x = \frac{\pi}{5}$ for some $x \in (-1, 1)$,then the value of $\cos^{-1} x$ is

$A$ possible value of $\tan \left(\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}\right)$ is :

The equation $\sin^{-1}x - \cos^{-1}x = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$ has

If $\sin ^{-1} \frac{x}{5}+\sin ^{-1} \frac{4}{5}=\frac{\pi}{2}$,then $x=$ . . . . . . .