Integer solutions of $2\cos\left(\frac{p\pi}n\right)+2\cos\left(\frac{q\pi}n\right)+4\cos\left(\frac{p\pi}n\right)\cos\left(\frac{q\pi}n\right)=1$

Question

I came across the following problem while trying to solve an eigenvalue problem. I want to know the integer solutions of the following equation

$$ \cos\left(\frac{p\pi}n\right) + \cos\left(\frac{q\pi}n\right) + 2 \cos\left(\frac{p\pi}n\right) \cos\left(\frac{q\pi}n\right) = \frac{1}{2}$$ where $p, q \in \{1, \dots, n-1\}$.

I have already observed that if $n = 6k$ for some $k \in \mathbb{N}$, then there is an immediate solution $p = 2k, q = 3k$. From numerical considerations, it seems that there are no solutions when $6 \not\mid n$. But, I do not have any explanation for it. Any help would be appreciated.

---

I also posted this question on MathOverflow, and received an answer that uses some advanced methods. I am looking forward for another solution that uses more elementary methods.