Recently, I formulated the following conjecture which seems novel.

**Conjecture**. For any positive odd integer $n$, we have the identity
$$\sum_{j,k=0}^{n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=\frac{n^2}2.\tag{1}$$

Using Galois theory, I see that the sum is a rational number. The identity $(1)$ has some equivalent versions, for example, $$\sum_{0\le j<k\le n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=\frac n4\left(n-(-1)^{(n-1)/2}\right)\tag{2}$$ and $$\sum_{1\le j<k\le n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=-\frac{n-(-1)^{(n-1)/2}}4\left(n+(-1)^{(n-1)/2}2\right).\tag{3}$$ It is easy to check $(1)$-$(3)$ numerically.

**Question.** How to prove the conjecture?

Your comments are welcome!