If `cos^(-1)x+cos^(-1)y+cos^(-1)=pi`, prove that `x^2+y^2+z^2+2x y z=1.`

Explanation

\begin{aligned}
&\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=\pi \\
&\cos ^{-1} x+\cos ^{-1} y=\pi-\cos ^{-1} z \\
&\cos ^{-1}\left(x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right)=\pi-\cos ^{-1} z \\
&x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}=\cos \left(\pi-\cos ^{-1} z\right) \\
&x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}=-\cos \left(\cos ^{-1} z\right) \\
&x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}=-z \\
&x y+z=\sqrt{1-x^{2}} \sqrt{1-y^{2}} \\
...