Given the expression defined for $x,y \in {[}0,2\pi{)} $ $$\begin{align} E(x,y)&=\cos(2x)+\cos(2y)+\cos(2x+2y) \\ &+4\cos{x}+4\cos y+4\cos(x+y)\\ &-2\cos(x-y)-2\cos(2x+y)-2\cos(2y+x) \end{align}$$ Wolfram-Alpha tells me that the minimum of this multivariable expression is $-\frac{27}{2}$, for $x=y=\frac{2\pi}{3}$.
How can this be proved (easily)?
I've tried using multivariable calculus but the calculations are really "messy". Using formulae that transform sums of trigonometric functions to products did not help either, in my case.