If $$\cot ^{-1} \sqrt{\cos \alpha}-\tan ^{-1} \sqrt{\cos \alpha}=x$$, then $$\sin x$$ is equal to

If $$x \in \left( {0,{\pi \over 2}} \right)$$, then the value of $${\cos ^{ - 1}}\left( {{7 \over 2}(1 + \cos 2x) + \sqrt {({{\sin }^2}x - 48{{\cos }^2}x)\sin x} } \right)$$ is equal to

The minimum value of $${({\sin ^{ - 1}}x)^3} + {({\cos ^{ - 1}}x)^3}$$ is equal to