karseme
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So, I've got an assignment to prove that f(x)=\cos{(n \cdot \arccos{x})} is a polynomial for \forall n \in \mathbb{N}. Also, we were suggested to use mathematical induction. So, I've tried:
Base step: n=1 \implies f(x)=\cos{(\arccos{x})}=x
Assumption step: f(x)=\cos{(n \cdot \arccos{x})}, \forall n \in \mathbb{N}
Induction step: 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.
Base step: n=1 \implies f(x)=\cos{(\arccos{x})}=x
Assumption step: f(x)=\cos{(n \cdot \arccos{x})}, \forall n \in \mathbb{N}
Induction step: 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.