Prove $\sin(\alpha/n)$ and $\cos(\alpha/n)$ are algebraic over $K=\mathbb{Q}(\sin\alpha)$ for any positive integer $n$.

Question

I observed that $\sin\alpha$ and $\cos\alpha$ are clearly algebraic over K. I'm not sure if I should use induction on this one because I would have to somehow deal with $\sin\frac{\alpha}{n-1}$ and $\cos\frac{\alpha}{n-1}$. In any case, I need to show that I can write $\sin(\alpha/n)=P(\sin\alpha,\cos\alpha)$ for $P \in \mathbb{Q}[X,Y]$ and the same with $\cos(\alpha/n)$. I was thinking of using the identity $$\sin(nx)=Im [(\cos x+i\sin x)^n]$$ and then let $x=\alpha/n$, but then I would have to deal with all the cosine terms, which I can't assume they're algebraic over $K$. What are useful identities that I can use here?

Answer 1

Hint:

use $$ 2i\sin x = e^x-e^{-x} \qquad 2\cos x= e^x+e^{-x} $$

Answer 2

Use just $\sin(x)^2 + \cos(x)^2 = 1$, there are two polynomials $P$ and $Q$ with integer coefficients that $$\sin(nx) = P(\sin(x))\cos(x) + Q(\sin(x))$$ Put $Q$ on the left side and raise it to square power. You obtain a polynomial with integer coefficient having $\sin(nx)$ and $\sin(x)$ as variables. And $\cos(x)$ is at most second order extension of $\mathbb{Q}(\sin(x))$, it is easy to conclude.