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

The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$,where $0 \leq \cos ^{-1} x \leq \pi$ and $-\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}$,is

Prove $\cos ^{-1} \frac{4}{5} + \cos ^{-1} \frac{12}{13} = \cos ^{-1} \frac{33}{65}$

The value of $\cot ^{-1}\left[\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right]$,where $x \in\left(0, \frac{\pi}{4}\right)$ is

If $2 \tan^{-1}(\cos x) = \tan^{-1}(2 \operatorname{cosec} x)$,then the value of $x$ is

If $\cot(\cos^{-1} x) = \sec(\tan^{-1} \frac{a}{\sqrt{b^2 - a^2}})$, then the value of $x$ is