If $y = \sin^{-1}\left(\frac{19}{20}x\right) + \cos^{-1}\left(\frac{19}{20}x\right)$,then $\frac{dy}{dx} = $

$2 \cos ^{-1} x = \sin ^{-1} \left( 2 x \sqrt{1 - x^2} \right)$ is valid for all values of $x$ satisfying

If $\sec ^{-1}\left(\frac{5}{x}\right)+\sin ^{-1} \left(\frac{4}{5}\right)=\frac{\pi}{2}$,where $x \neq 0$,then $x=$ . . . . . . .

If $\sec^{-1} x = \csc^{-1} y$,then $\cos^{-1} \frac{1}{x} + \cos^{-1} \frac{1}{y} = $

Evaluate: $\cos(\cos^{-1}(-\frac{1}{2}) + \frac{\pi}{3}) = $

If ${\sin ^{ - 1}}\left( {x - \frac{{{x^2}}}{2} + \frac{{{x^3}}}{4} - \dots} \right) + {\cos ^{ - 1}}\left( {{x^2} - \frac{{{x^4}}}{2} + \frac{{{x^6}}}{4} - \dots} \right) = \frac{\pi }{2}$ for $0 < |x| < \sqrt 2$,then $x$ equals