$\cos \left(2 \left(\tan ^{-1} \frac{1}{5}+\tan ^{-1} 5\right)\right) = $ . . . . . .

The value of $\tan ^{-1}\left\{\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right\}+\frac{1}{2} \cos ^{-1} x$ is

If $2 \sin^{-1} x = \sin^{-1}(2x \sqrt{1-x^2})$,then $x \in$ . . . . . . .

If $\sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2}$,then the value of $x^{100} + y^{100} + z^{100} - \frac{9}{x^{101} + y^{101} + z^{101}}$ is equal to

If $\cos ^{-1} 2x + \cos ^{-1} 3x = \frac{\pi}{3}$, then $x =$

If $\alpha = \cos^{-1}\left(\frac{3}{5}\right)$ and $\beta = \tan^{-1}\left(\frac{1}{3}\right)$,where $0 < \alpha, \beta < \frac{\pi}{2}$,then $\alpha - \beta$ is equal to