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.

$cos x +cos y -cos(x+y)=2\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})-(2\cos^2(\frac{x+y}{2})-1)=2\cos(\frac{x+y}{2})\cdot\left(\cos(\frac{x-y}{2})-cos(\frac{x+y}{2})\right)+1\le $
$2\cos(\frac{x+y}{2})\cdot\left(1-cos(\frac{x+y}{2})\right)+1\le 1/2+1=\frac{3}{2}$, where the inequality is by $ab\le (\frac{a+b}{2})^2$, so $\cos(\frac{x+y}{2})\cdot\left(1-cos(\frac{x+y}{2})\right)\le 1/4$ . The equality holds iff $x=y=\pi/3$.
