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 $$\Theta(x,y)\le \Theta(x,z)+\Theta(z,y) .$$  

The inequality simply says that the "angle between $\vec x$ and $\vec z$ plus the angle between $\vec z$ and $\vec y$ is never smaller than the angle between $\vec x$ and $\vec y$". That's obvious in the Euclidean space. First, as ncmathsadist pointed out, the angles don't change if you normalize $\vec x,\vec y,\vec z$ to unit vectors. 
Then the angles (in radians) between two vectors are measured just as distances measured along the surface of the unit sphere and the inequality says that there the crow-fly, straight distance between $x$ and $y$ can't be made shorter by inserting a detour to $z$. Well, I hope that it's not hard to see how to prove that the crow-fly distance is the shortest path between the two vectors on the sphere.
