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Prove that $x = \cos(\frac{2\pi}{n})$ or $\cos(\frac{4\pi}{n})$ is a root of a polynomial of degree $n$

Prove that $x = \cos(\frac{2\pi}{n})$ or $\cos(\frac{4\pi}{n})$ is a root of a polynomial of degree $n$. Where $n$ is a positive integer.

I’ve looked up the cyclotomic polynomials, and the answer is given to be $x^n+x^{n-2}+…+x+1$ when $n$ is odd and $x^n-x^{n-2}+…-x+1$ when $n$ is even.

I’m not quite sure how to go about proving this. I know that a root of a polynomial can be found by taking the derivative of that polynomial, but this makes it difficult to figure out a way to generalize it.

A:

For simplicity, we will suppose that $n$ is odd, the case that $n$ is even can be worked out similarly.

We have

$$f(x)=x^n-\cos(2\pi/n)-1=0$$

By taking the derivative of $f(x)$ we get

$$\frac{d}{dx}f(x)=nx^{n-1}-2\sin(2\pi/n)=0$$

Which implies that $x=\cos(\frac{2\pi}{n})$.

Hope this helps.

#pragma once

#include

namespace opensubdiv {

inline std::vector Split(Algorithm* algo, Vector4i size, Vector4f center) {

const size_t nc = algo->numChildren;

size_t size_1 = std::min(size[0], size[1]);

size

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