In a triangle PQR with usual notations, $\angle \mathrm{R}=\frac{\pi}{2}$. If $\tan \frac{\mathrm{P}}{2}$ and $\tan \frac{\mathrm{Q}}{2}$ are the roots of the equation $a x^2+b x+c=0(a \neq 0)$, then

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

Considering only the principal values of the inverse trigonometric functions, the value of $\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)$ is

If the area bounded by the curve $x^2=4 y, \mathrm{X}$-axis and the line $x=4$ is divided into equal areas by the line $x=\alpha$, then the value of $\alpha$ is …