Consider a polynomial with roots $\cos^2 \left(\frac{j \pi}{2n + 1}\right), 1 \leq j \leq n$. The coefficients of this polynomial are sums of these numbers taken one at a time, two at a time, three at a time, and so on, up to sign. That is
$$ \sum\limits_{j = 1}^n \cos^2 \left( \frac{j\pi}{2n+1} \right), \sum\limits_{j<k; \;j,k = 1}^n \cos^2 \left(\frac{j\pi}{2n+1} \right)\cos^2 \left( \frac{k\pi}{2n+1} \right), $$
and so on. I can prove that $\sum\limits_{j = 1}^n \cos^2 \left( \frac{j\pi}{2n+1} \right) = \frac{n}{2} - \frac{1}{4}$. Upon examination, other sums of this form also appear to be rational numbers. I would like to know if there is a expression for these sums as a function of $n$.
If so, what is it or how can I find and prove it?
