If $\sin^{-1}(x)+\sin^{-1}(y)+\sin^{-1}(z)=\pi$ then prove that $x*\sqrt{1-x^2}+y*\sqrt{1-y^2}+z*\sqrt{1-y^2}=2xyz$

Question

I assumed $\sin^{-1}(x)=A$, $\sin^{-1}(y)=B$ and $\sin^{-1}(z)=C$.

Then it went the following way

$ A + B + C = \pi$

$A + B = \pi - C$

$\sin(A+B)= \sin(\pi-C)$

$\sin A \cos B + \cos A \sin B = sin C$

$x\sqrt{1-y^2}+y\sqrt{1-x^2} = \sin(\sin^{-1}(z))$

Am i right in doing so or heading wrong?