Consider the following input
Assuming[Element[y, Integers], (-1 + E^(I \[Pi] (x + y)))/(-1 + E^(I \[Pi] (x - y)))//FullSimplify]
(-1 + E^(I \[Pi] (x + y)))/(-1 + E^(I \[Pi] (x - y)))
the output comes out exactly the same as the input. However, note that for integer y
we have
$$e^{i\pi(x+y)}=e^{i\pi(x+2y-y)}=\underbrace{e^{2\pi i y}}_{=1}e^{i\pi(x-y)}=e^{i\pi(x-y)}$$
With this I would expect the above mathematica input to return 1
instead of unaltered input.
How to properly simplify this in mathematica?