We have,
sin-1 x + sin-1 y + sin-1 z = π
sin-1 x + sin-1 y = π - sin-1 z
\(sin^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^ 2}]=\pi-sin^ {-1}z\)
\((x\sqrt{1-y^2}+y\sqrt{1-x^2})=sin(\pi -sin^{-1}z)=sin(sin^{-1}z)\)
\((\because sin(\pi-\theta)=sin\theta)\)
\(x\sqrt{1-y^2}+y\sqrt{1-x^2}=z\)
\(x\sqrt{1-y^2}=z-y\sqrt{1-x^2}\)
On squaring both sides we get
\(x^2(1-y^2)=z^2+y^2(1-x^2)-2yz \sqrt{1-x^ 2}\)
\(=z^2+y^2-x^2y^2-2yz\sqrt{1-x^2}\)
\(x^ 2-y^2-z^ 2+2yz\sqrt{1-x^2}=0\)
