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

The solution of $\sin^{-1} x - \sin^{-1} 2x = \pm \frac{\pi}{3}$ is

If $\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2},$ then the value of $x$ is

The value of ${\tan ^{ - 1}}\left[ {\cos \left( {2\,{{\tan }^{ - 1}}\frac{3}{4}} \right)\, + \,\sin \,\left( {2\,{{\cot }^{ - 1}}\frac{1}{2}} \right)} \right]$ is

If $4 \sin ^{-1} x + \cos ^{-1} x = \pi$,then $x = $

Show that $\sin ^{-1}(2 x \sqrt{1-x^{2}})=2 \cos ^{-1} x$ for $\frac{1}{\sqrt{2}} \leq x \leq 1$.