If cos⁻¹ x + cos⁻¹ y + cos⁻¹ z = π, then find the value of x² + y² + z² + 2xyz.

Let $\cos^{-1}x=A$, $\cos^{-1}y=B$, $\cos^{-1}z=C$, so $x=\cos A$, $y=\cos B$, $z=\cos C$ and $A+B+C=\pi$. Then $C=\pi-(A+B)$, so $z=-\cos(A+B)=-(xy-\sqrt{1-x^2}\sqrt{1-y^2})$. Hence $z+xy=\sqrt{(1-x^2)(1-y^2)}$. Squaring: $z^2+2xyz+x^2y^2=1-x^2-y^2+x^2y^2$. Therefore $x^2+y^2+z^2+2xyz=1$.