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

If $\cos ^{-1}\left(\frac{12}{13}\right)+\sin ^{-1}\left(\frac{3}{5}\right)=\sin ^{-1} P$,then the value of $P$ is

Prove $\tan ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)=\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} x$,where $-\frac{1}{\sqrt{2}} \leq x \leq 1$.

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

The value of $\tan \left( \tan^{-1} \frac{1}{2} - \tan^{-1} \frac{1}{3} \right)$ is

If $\sin^{-1} \frac{1}{3} + \sin^{-1} \frac{2}{3} = \sin^{-1} x$,then $x$ is equal to