If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=\pi$ and $x^2+y^2+z^2+k x y z=1$,then $k$ is

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 $8(\alpha+\beta)=$

If $(\tan ^{-1} x)^2+(\cot ^{-1} x)^2=\frac{5 \pi^2}{8}$,then the value of $x$ is

The derivative of $\cos ^{-1}\left(2 x^{2}-1\right)$ with respect to $\cos ^{-1} x$ is

If $\cos ^{-1} 2x + \cos ^{-1} 3x = \frac{\pi}{3}$ and $4x^2 = \frac{a}{b}$,then $a + b =$

Prove that $\tan ^{-1} x+\tan ^{-1} \frac{2 x}{1-x^{2}}=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)$,where $|x| < \frac{1}{\sqrt{3}}$.