# How to assume a number to be a half integer?

For integer x $$\left(\cos 2\pi x\right) ^2=1$$ and I can check this as follows

Simplify[Cos[2 \[Pi] x]^2, Assumptions -> x \[Element] Integers] 

and get 1 as an output. The same holds when x is half-integer. How do I assume this in Mathematica? More generally, is there a nice way to implement assumptions of the type say $$x\in 3\mathbb{Z}+2$$ without redefining variables that I use?