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

If $k \le \sin^{-1}x + \cos^{-1}x + \tan^{-1}x \le K$,then

$\tan \left[ {\frac{1}{2}{{\sin }^{ - 1}}\left( {\frac{{2a}}{{1 + {a^2}}}} \right) + \frac{1}{2}{{\cos }^{ - 1}}\left( {\frac{{1 - {a^2}}}{{1 + {a^2}}}} \right)} \right] = $

If $\cos ^{-1} x - \cos ^{-1} \frac{y}{2} = \alpha$,where $-1 \leq x \leq 1, -2 \leq y \leq 2, x \leq \frac{y}{2}$,then for all $x, y$,$4x^2 - 4xy \cos \alpha + y^2$ is equal to

If $y = \tan^{-1}\left( \frac{\sqrt{a} - \sqrt{x}}{1 + \sqrt{ax}} \right)$,then $\frac{dy}{dx} = $

If $\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2},$ then the value of $x$ is