The value of $\sin \left(\cos ^{-1}\left(-\frac{1}{3}\right)-\sin ^{-1}\left(\frac{1}{3}\right)\right)$ is

If $y = \tan^{-1} \frac{x}{1+2x^2} + \tan^{-1} \frac{x}{1+6x^2} + \tan^{-1} \frac{x}{1+12x^2}$,then $\left(\frac{dy}{dx}\right)_{x=\frac{1}{2}} = $

Prove $3 \sin ^{-1} x = \sin ^{-1}(3 x - 4 x^{3})$,where $x \in [-\frac{1}{2}, \frac{1}{2}]$.

The value of $\tan \left\{\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right\}$ is

If $a=\sin ^{-1}(\sin (5))$ and $b=\cos ^{-1}(\cos (5))$,then $a^2+b^2$ is equal to

If $\sin ^{-1}\left(x-\frac{x^2}{2}+\frac{x^3}{4}-\ldots \infty\right) + \cos ^{-1}\left(x^2-\frac{x^4}{2}+\frac{x^6}{4}-\ldots \infty\right)=\frac{\pi}{2}$ and $0 < x < \sqrt{2}$,then $x$ is equal to