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

Find $\frac{dy}{dx}$ if $y = \sin^{-1}(2x\sqrt{1-x^2})$ where $-\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}$.

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-$2$: ${\tan ^{ - 1}}\left[ {\frac{{1 + \log {x^2}}}{{1 - \log {x^2}}}} \right] = {\tan ^{ - 1}}1 + {\tan ^{ - 1}}(\log {x^2})$

$\cos \left[\sin ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)\right]=$

$\begin{aligned} & \text{If } \cot \left(\cos ^{-1} x\right)=\sec \left\{\tan ^{-1}\left(\frac{a}{\sqrt{b^2-a^2}}\right)\right\} \\ & b>a, \text{ then } x= \end{aligned}$

If $a < \frac{1}{32},$ then the number of solutions of $(\sin^{-1} x)^3 + (\cos^{-1} x)^3 = a\pi^3$ is