So, I've got an assignment to prove that $ \displaystyle f(x)=\cos{(n \cdot \arccos{x})} $ is a polynomial for $ \displaystyle \forall n \in \mathbb{N} $. Also, we were suggested to use mathematical induction. So, I've tried:

Base step: $ \displaystyle n=1 \implies f(x)=\cos{(\arccos{x})}=x$

Assumption step: $ \displaystyle f(x)=\cos{(n \cdot \arccos{x})}, \forall n \in \mathbb{N} $

Induction step: $ \displaystyle f(x)=\cos{((n+1) \cdot \arccos{x})}=\cos{(n \arccos{x}+\arccos{x})}=\cos{(n \arccos{x})}\cos{( \arccos{x})}-\sin{(n \arccos{x})}\sin{( \arccos{x})}=f(x) \cdot x -\sin{(n \arccos{x})}\sin{( \arccos{x})}$

And I don't know what to do with sine.