If $\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1$,then the value of $x$ is

If $\tan ^{-1}\left[\frac{1}{1+1 \cdot 2}\right]+\tan ^{-1}\left[\frac{1}{1+2 \cdot 3}\right]+\cdots+\tan ^{-1}\left[\frac{1}{1+n(n+1)}\right]=\tan ^{-1}[x]$,then $x=$

Find the value of $\cot \left(\tan ^{-1} a+\cot ^{-1} a\right)$.

$\tan ^{-1}\left(\frac{1}{2 \sqrt{2}}\right)+\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)=\cos ^{-1} x$, then $x=$

If $y = \cot^{-1} \left( \frac{1 + \sin 5x}{\cos 5x} \right)$, then the value of $\frac{dy}{dx}$ is

$\sin^{-1} \sqrt{\frac{x}{x+a}}$ is equal to