AN INTERESTING RELATION
I just copy here an answer of mine of this post, just in case OP decided to delete it. Link: https://math.stackexchange.com/q/5005043/1231540 First we state two results: - **Proposition 1:** Let $n \ge 3$. The minimal polynomial of $\cos(2\pi/n)$ - denoted by $\psi_n$ - has degree $\phi(n)/2.$ Here $\phi$ is Euler totient function. - **Proposition 2:** The roots of $\psi_n$ are $\cos(2k\pi/n)$ where $\gcd(k,n)=1$ and $0 \le k \le \lfloor \frac{n}{2}\rfloor$. The proofs of the above two propositions can be found in this paper: https://www.jstor.org/stable/2324301 ***Proof of the first identity:*** It can be observed that $4\sin^2(\pi/n) = 2-4\cos(2\pi/n)$, so the polynomial $\Phi_n$ and $\psi_n$ have the same degree, which is $\phi(n)/2$. In particular, $\Phi_{2m}$ has degree $\phi(2m)/2$. If $m$ is odd, we clearly have $\phi(2m) = \phi(m)$, which means $\Phi_{2m}$ and $\Phi_m$ have the same degrees. By definition, $\Phi_{2m}(x)$ is irreducible and has...
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