How do you find the sum for this term $$ \cos^{n}(\pi x) + \cos^{n}(\pi y) $$
if $x-y=a$, where $a\in N^{+}$ and $x,y \in R$
Note: for $n=1$
$$ \cos(\pi x) + \cos(\pi y)=0 $$
by use property
$$ \cos(\alpha) + \cos(\beta)= 2\cos(\frac{\alpha+\beta}{2})\cos(\frac{\alpha-\beta}{2}) $$
$$ \cos^{n}(\pi x) + \cos^{n}(\pi y) $$ $$=\cos^n(\pi y + \pi a)+\cos^n(\pi y)$$
Notice that $$\cos(\theta+ \pi)=-\cos(\theta)$$ and $$\cos(\theta+2\pi)=\cos(\theta)$$
Thus if $a$ and $n$ are both odd integers, the result is $0$; otherwise, the result is $2\cos^n(\pi y)$