If $\tan ^{-1} x+\tan ^{-1} y=\frac{4 \pi}{5}$,then $\cot ^{-1} x+\cot ^{-1} y$ is equal to

If $\sin ^{-1} x - \cos ^{-1} x = \frac{\pi}{6}$,then $x$ is equal to

If $\alpha$ and $\beta$ are the least and the greatest values of $f(x)=(\sin ^{-1} x)^2+(\cos ^{-1} x)^2$ for all $x \in [-1, 1]$ respectively,then $8(\alpha+\beta)=$

The differential coefficient of ${\tan ^{ - 1}}\left( \frac{{2x}}{{1 - {x^2}}} \right)$ with respect to ${\sin ^{ - 1}}\left( \frac{{2x}}{{1 + {x^2}}} \right)$ is

If $\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3 \pi}{2}$,then $x^{100}+y^{100}+z^{100}=$

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