If $\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x$, then the value of $\sin x$ is

If $[x]^2-5[x]+6=0$, where $[\cdot]$ denotes the greatest integer function, then

The value of the integral $\int_0^{\frac{\pi}{2}} \frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} \mathrm{dx}$ is

Variance of first n natural numbers is ________.