I am trying to calculate the minimal polynomials of $h_{1}=-\cos(\pi/n)-\sqrt{\cos(2\pi/n)}$ and $h_{2}=-\cos(\pi/n)+\sqrt{\cos(2\pi/n)}$ when $n$ is odd. I think (and numerical calculations suggest that) these two have the same minimal polynomial.

I tried the calculation $(x-h_{1})(x-h_{2})=x^{2}+2x\cos(\pi/n)+\sin^{2}(\pi/n)$ which upon simplification is just $$4x^{2}-4\left(\zeta_{n}^{n-1}+\zeta_{n}^{n+1}\right)x+\left(\zeta_{n}^{n-2}+2+\zeta_{n}^{n+2}\right)$$ where $\zeta_{n}=(-1)^{1/n.}$

So the minimal polynomial $p_{n}(x)$ will have the above quadratic as a factor in $C[x]$. How do we get these minimal polynomials?

Calculations in mathematica suggest that the minimal polynomials are non-trivial factors of polynomials generated by the rational function
$$J(x,t):=\frac{1+(4x-1)t+(4x-1)t^{2}+t^{3}}{1-4(1+x^{2})t+6(1+4x^{2})t^{2}-4(1+x^{2})t^{3}+t^{4}}$$