How do I prove equality of x and y? If $0\leq x,y\leq\frac{\pi}{2}$ and $\cos x +\cos y -\cos(x+y)=\frac{3}{2}$, then how can I prove that $x=y=\frac{\pi}{3}$?
Your help is appreciated.I tried various formulas but nothing is working.

You could also attempt an geometric proof. First, without loss of generality you can assume
$0 <x,y < \frac{\pi}{2}$.
Construct a triangle with angles $x,y, \pi-x-y$.
Let $a,b,c$ be the edges. Then by cos law, you know that 
$$\frac{a^2+b^2-c^2}{2ab}+ \frac{a^2+c^2-b^2}{2ac}+ \frac{b^2+c^2-a^2}{2bc}=\frac{3}{2}$$
and you need to show that $a=b=c$.
The inequality is
$$c(a^2+b^2-c^2)+b(a^2+c^2-b^2)+a(b^2+c^2-a^2)=3abc$$
or 
$$a^2b+ab^2+ac^2+a^2c+bc^2+b^2c=a^3+b^3+c^3+3abc \,.$$
This should be easy to factor, but I fail to see it....
