We have the function:

$\displaystyle ArcCos[\frac{x^2 + y^2 - z^2}{2xy}] + ArcCos[\frac{y^2 + z^2 - x^2}{2yz}] + ArcCos[\frac{z^2 + x^2 - y^2}{2zx}] $

defined for: $\displaystyle 0<x<=y<=z<x+y$

It is clear that the function is defined for all values of x, y and z that satisfy the conditions above. Moreover it can be verified that it is constant and its value equals pi.

I need to find the total differential of the function (I know it must be null, since the function itself is constant, but I still need to prove this). However, it's just too complicated, differentiating with respect to each variable: the expression is far too hard to handle.

Can anyone come up with a smart substitution that should make the total differential easier to work on, and prove finally that it is 0? Thanks guys.