sequences and series - Calculate the limit of $a_{n+1}=cospi a_n$

This is my first time to ask a question here, so I'm sorry if I make any mistake.

I'm trying to calculate the limit of $a_n$, which is defined as $a_{n+1}:=\cos\pi a_n$.

The first term is not given in this problem, so I have to prove it by cases.
I am badly stuck not knowing how to find the limit of this sequence and where to start the proof. After I substituted some values of $a1$ and calculate it, I found the convergence value might be $-1$ with almost every $a1$ (there were different ones), but I end up stuck here.