What does this notation mean:
To provide some context, here are some of the exercises related to it:
I initially thought the notation was such that 
$cos(2pi/n^n)+sin(2pi/n^n)$.
This doesn't quite seem like the proper interpretation to me.
I noticed that since $x=cos(t), y=sin(t)$ would graph a circle in the xy-plane, $x^2+y^2$ will always be equal to $1$, I thought perhaps this was related to the direction we should be heading in for this exercise.
However that line of logic breaks down when I look at the following exercises.
This has nothing to do with the zeta function, and in short, you are overthinking the notation. You can choose any variable you like if you would prefer. I'll use $\alpha$.
So fix an $n$, and let $\alpha = \cos(2\pi/n) + i\sin(2\pi/n)$, which also happens to be $e^{2\pi i/n}$. When we say $\alpha^n$, we mean "raise $\alpha$ to the $n$th power", like we usually mean when we write an exponent. So $\alpha^n = \left( \cos (\frac{2\pi}{n} ) + i\sin (\frac{2\pi}{n} )\right)^n$, which you should show is also equal to $1$.