Just a starter

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

$\cos^{-1}(xy - \sqrt{1 - x^2}\sqrt{1 - y^2})$ $=$ $\pi\ - \cos^{-1}z$

$xy - \sqrt{1 - x^2}\sqrt{1 - y^2}$ $=$ $\cos (\pi - \cos^{-1}z)$

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

$xy - \sqrt{1 - x^2}\sqrt{1 - y^2}\ = - z$

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

$x^2y^2 + 2xyz + z^2 = 1 - x^2 + x^2y^2 - y^2$

$\boxed{x^2 + y^2 + z^2 + 2xyz = 1}$