Consider the integral
$$\int_{-\pi}^\pi \int_{-\pi}^\pi \,dk \,d\kappa \dfrac{\sin(k a) \sin(k b) \sin(\kappa b) \sin(\kappa c)}{\cos(k)-\cos(\kappa)+i \alpha} $$ where $x,y,z$ are arbitrary real numbers and $\alpha > 0 $. If we only had one integral to do, then the result would be simply. We would simple let $z = e^{i \kappa z}$ and use the residue theorem on the poles inside the unit circle. But the roots, will contain factors of $\sqrt{(\cos(k)+i \alpha)^2-1}$, including in the denominator. So if we want to use the residue theorem again, we'd have to worry about the branch cuts of the square root, leading to something very complicated.
Is there something that I'm missing here? Perhaps a change of variables to make things easier? In any case, I can't see things simplifying. Thanks!