Can you prove, that the integral is zero or that this function is odd

Question

The problem is to prove that: $\int\limits_{-\pi}^{\pi}\int\limits_{-\pi}^{\pi} (a\sin(x)+b\cos(x)+c\sin(y)+d\cos(y))e^{-2\cos(y-x)+a\cos(x)-b\sin(x)+c\cos(y)-d\sin(y)}dxdy$=0

I have an idea to represent a function under the integral like the odd function, but I can't.

Also, I can prove, that: $\int\limits_{-\pi}^{\pi} (a\cos(x)+b\sin(x))e^{a\sin(x)-b\cos(x)}\,dx=0$

Maybe it helps you.

Answer 1

Writing $e(x,y)=e^{2\cos(x-y)+a\cos x-b\sin(x)+c\cos y-d\sin y}$ we have \begin{equation*} \int\limits_{-\pi}^{\pi}\int\limits_{-\pi}^{\pi} (a\sin(x)+b\cos(x)+c\sin(y)+d\cos(y))e(x,y)dxdy=-\int\limits_{-\pi}^{\pi}\int\limits_{-\pi}^{\pi} (\partial_x+\partial_y)e(x,y)dxdy. \end{equation*} The result follows.