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

Prove that $\sin ^{-1} \frac{8}{17}+\sin ^{-1} \frac{3}{5}=\tan ^{-1} \frac{77}{36}$

If $y = \sin^{-1}\left(\frac{x^2 - 1}{x^2 + 1}\right) + \sec^{-1}\left(\frac{x^2 + 1}{x^2 - 1}\right)$,$|x| > 1$,then $\frac{dy}{dx}$ is equal to:

$\tan \left[ {\frac{\pi }{4} + \frac{1}{2}{{\cos }^{ - 1}}\frac{a}{b}} \right] + \tan \left[ {\frac{\pi }{4} - \frac{1}{2}{{\cos }^{ - 1}}\frac{a}{b}} \right] = $

$A$ value of $x$ satisfying the equation $\sin \left[ \cot^{-1} (1 + x) \right] = \cos \left[ \tan^{-1} x \right]$ is

If $\sin^{-1} \frac{1}{3} + \sin^{-1} \frac{2}{3} = \sin^{-1} x$,then $x$ is equal to