If ${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\pi$,  then  $x\sqrt{1-x^2}+y\sqrt{1-y^2}+z\sqrt{1-z^2}=$ 

Given  ${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\pi$

Let ${\sin }^{-1}x=A\Rightarrow \sin A=x,{\sin }^{-1}y=B\Rightarrow \sin B=y,{\sin }^{-1}z=C\Rightarrow \sin C=z$

$\begin{array}{l}A+B+C=\pi \Rightarrow \sin 2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C\\ \Rightarrow 2\sin A\cos A+2\sin B\cos B+2\sin C\cos C=4\sin A\sin B\sin C\\ \Rightarrow x\sqrt{1-x^2}+y\sqrt{1-y^2}+z\sqrt{1-z^2}=2xyz\end{array}$