Q:

Why Refine does not simplify Cos[2*Pi*FractionalPart[1/2*(i + j + k)]] to $(-1)^{i+j+k}$?

I am obtaining in Mathematica:
Cos[2*Pi*FractionalPart[1/2*(i + j + k)]]

We know that for $i, j, k$ being positive integers this expression after simplification should give:
$$
(-1)^{i+j+k}
$$
I would like to know why the one of the solutions proposed for this similar question:
 Refine[Cos[2*Pi*FractionalPart[1/2*(i + j + k)]], 
    Assumptions -> {Element[{i, j, k}, Integers], i > 0, j > 0, k > 0}]

did not work for this case?

A:

One can use x-Floor[x] instead of FractionalPart[x] for positive x
FullSimplify[Cos[2 Pi ((i + j + k)/2 - Floor[(i + j + k)/2])], 
   Assumptions -> (i | j | k) ∈ Integers]

(-1)^(i + j + k)

