Would you please help me solve Exercise 2, which I repeat here:
Suppose that $p$ and $q$ are relatively-prime positive integers. Show that if $\cos p \alpha$ and $\cos q \alpha$ are rational, then $\cos \alpha$ is rational or $\alpha$ is a multiple of $\pi / 6$.
The exercise is a supplement to An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery.
There are two approaches to solve the problem. The first is to assume that $\cos \alpha$ is irrational and prove that $\alpha$ must then be a multiple of $\pi / 6$. In this case, I know from Exercise 1 that $\cos (p + q) \alpha$ is irrational, which is equivalent to the fact that $\sin p \alpha \sin q \alpha$ is irrational, deduced from my solution to Exercise 1. I can construct examples that satisfy this approach but cannot devise a general proof.
The other approach is to assume that $\alpha$ is not a multiple of $\pi / 6$ and prove that $\cos \alpha$ must then be rational. Once again, I can construct examples but cannot figure out how to prove it in general.