How to prove the inequality $\Theta(x,y)\le \Theta(x,z)+\Theta(z,y)$? Let $x, y$ be two complex vectors, $$\cos\Theta(x,y):=\operatorname{Re} \frac{y^*x}{\|x\|\|y\|} .$$ Then I want to prove that $$\Theta(x,y)\le \Theta(x,z)+\Theta(z,y) .$$  

You should think about distances on the unit sphere.
