If $\sin^{-1} x + \cos^{-1} y = \frac{2\pi}{5}$,then $\cos^{-1} x + \sin^{-1} y$ is

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

$\sin^{-1} \frac{1}{\sqrt{5}} + \cot^{-1} 3$ is equal to

If ${x_1}, {x_2}, {x_3}, {x_4}$ are roots of the equation ${x^4} - {x^3}\sin 2\beta + {x^2}\cos 2\beta - x\cos \beta - \sin \beta = 0$,then ${\tan ^{ - 1}}{x_1} + {\tan ^{ - 1}}{x_2} + {\tan ^{ - 1}}{x_3} + {\tan ^{ - 1}}{x_4} = $

If $\frac{\sin ^{-1} x}{a}=\frac{\cos ^{-1} x}{b}=\frac{\tan ^{-1} y}{c}$ and $0 < x < 1,$ then the value of $\cos \left(\frac{\pi c}{a + b}\right)$ is

The derivative of $\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $x$ is: