Can it be Shown That an Angle $+ 2\pi$ = the Same Angle?

Question

This might be some kind of foundational assertion. Given any angle $\psi$ and $x\in\Bbb Z^+$ I'd like to see proof that:

$$\psi = \psi + 2x\pi$$

It's hard cause numerically this isn't true, but in the sense of angles it is. I specifically need this to make my proof based on more than my on personal assertion here: https://math.stackexchange.com/a/2144499/194115

Answer 1

Mathematically, it is not true that $\psi = \psi + 2x\pi$.

What is true is that, for any trig function, since they have a period of $2\pi$ (some of them, like $\tan$, $\pi$), that $f(\psi) = f(\psi + 2x\pi)$.

Answer 2

You could define an equivalence relation on the set of angles: $\alpha \sim \beta$ if $\alpha = \beta + 2\pi n$ for some integer $n$