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

The value of $x$ for which $\sin(\cot^{-1} (1 + x)) = \cos(\tan^{-1} x)$ is

$\tan \left[ {\frac{1}{2}{{\sin }^{ - 1}}\left( {\frac{{2a}}{{1 + {a^2}}}} \right) + \frac{1}{2}{{\cos }^{ - 1}}\left( {\frac{{1 - {a^2}}}{{1 + {a^2}}}} \right)} \right] = $

Solve for $x$: $\tan^{-1}\left(\frac{1-x}{1+x}\right) = \frac{1}{2} \tan^{-1} x$,where $x > 0$.

If $y = \sec^{-1}\left( \frac{x + 1}{x - 1} \right) + \sin^{-1}\left( \frac{x - 1}{x + 1} \right)$,then $\frac{dy}{dx} = $

Prove that $\tan ^{-1} \sqrt{x} = \frac{1}{2} \cos ^{-1} \left( \frac{1-x}{1+x} \right)$,where $x \in [0, 1]$.