The value of $\tan \left[2 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi}{4}\right]$ is

If $y = \cos^{-1}(\cos x)$,then find $\frac{dy}{dx}$ at $x = \frac{5\pi}{4}$.

If $\sin ^{-1}\left(x-\frac{x^2}{2}+\frac{x^3}{4}-\ldots \infty\right) + \cos ^{-1}\left(x^2-\frac{x^4}{2}+\frac{x^6}{4}-\ldots \infty\right)=\frac{\pi}{2}$ and $0 < x < \sqrt{2}$,then $x$ is equal to

If $y = \operatorname{cosec}^{-1}\left[\frac{\sqrt{x}+1}{\sqrt{x}-1}\right] + \cos^{-1}\left[\frac{\sqrt{x}-1}{\sqrt{x}+1}\right]$,then $\frac{dy}{dx} = $

$\cot ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) = $ . . . . . .

The value of ${\tan ^{ - 1}}\left[ {\cos \left( {2\,{{\tan }^{ - 1}}\frac{3}{4}} \right)\, + \,\sin \,\left( {2\,{{\cot }^{ - 1}}\frac{1}{2}} \right)} \right]$ is