Prove that f(x)=cos(narccos(x)) is polynomial

[karseme]

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.

 

[I like Serena]

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.

Hi karseme! ;)

How about assuming that $\sin{(n \arccos{x})}\sin{( \arccos{x})}$ is a polynomial in $x$? (Wondering)

\begin{aligned}\sin{((n+1) \arccos{x})}\sin{(\arccos{x})}
&= \big(\sin{(n \arccos{x})}\cos{(\arccos{x})} + \cos{(n \arccos{x})}\sin{(\arccos{x})} \big)\sin{( \arccos{x})} \\
&= x \sin{(n \arccos{x})}\sin{( \arccos{x})} + \cos{(n \arccos{x})}\sin^2{(\arccos{x})} \\
&= x \sin{(n \arccos{x})}\sin{( \arccos{x})} + \cos{(n \arccos{x})}(1 - \cos^2{(\arccos{x})})
\end{aligned}

 

[karseme]

Hi karseme! ;)

How about assuming that $\sin{(n \arccos{x})}\sin{( \arccos{x})}$ is a polynomial in $x$? (Wondering)

\begin{aligned}\sin{((n+1) \arccos{x})}\sin{(\arccos{x})}
&= \big(\sin{(n \arccos{x})}\cos{(\arccos{x})} + \cos{(n \arccos{x})}\sin{(\arccos{x})} \big)\sin{( \arccos{x})} \\
&= x \sin{(n \arccos{x})}\sin{( \arccos{x})} + \cos{(n \arccos{x})}\sin^2{(\arccos{x})} \\
&= x \sin{(n \arccos{x})}\sin{( \arccos{x})} + \cos{(n \arccos{x})}(1 - \cos^2{(\arccos{x})})
\end{aligned}

But, how can we assume that? It's like let's assume that every number is divisible by 3 for the sake of the convenicence. I don't see how can we assume that here. Maybe it is polynomial, I don't know. Anyway what can we achieve by assuming that, you're still left with $ x \sin{(n \arccos{x})}\sin{( \arccos{x})}$. And who says that $\sin{(n \arccos{x})}\sin{( \arccos{x})}$ is a polynomial. How to prove that.

 

[I like Serena]

But, how can we assume that? It's like let's assume that every number is divisible by 3 for the sake of the convenicence. I don't see how can we assume that here. Maybe it is polynomial, I don't know. Anyway what can we achieve by assuming that, you're still left with $ x \sin{(n \arccos{x})}\sin{( \arccos{x})}$. And who says that $\sin{(n \arccos{x})}\sin{( \ar
ccos{x})}$ is a polynomial. How to prove that.

Let's revise the induction hypothesis.
Let's make it: $f_n(x)=\cos(n \arccos x)$ is polynomial AND $g_n(x)=\sin(n \arccos x)\sin(\arccos x)$ is polynomial.
Is it true for a base case?
What will the induction step be?

 

[Greg]

$$\cos[(n+1)\arccos(x)]=\cos(n\arccos(x))\cos(\arccos(x))-\sin(n\arccos(x))\sin(\arccos(x))$$

$$=x\cos(n\arccos(x))+\frac{\cos[(n+1)\arccos(x)]-\cos[(n-1)\arccos(x)]}{2}$$

$$\cos[(n+1)\arccos(x)]=2x\cos(n\arccos(x))-\cos[(n-1)\arccos(x)]$$

I believe that's sufficient.