Find the minimal polynomial for $\cos(\frac{2\pi}{5})$ and $\sin(\frac{2\pi}{5})$ Let $\omega$ be the primitive 5th root of $1$, then $\cos(\frac{2\pi}{5}) = \frac{w+w^{-1}}{2}$ and $\sin(\frac{2\pi}{5}) = \frac{w-w^{-1}}{2i}$. How to find the minimal polynomial of  $\frac{w+w^{-1}}{2}$ then? (without using the Chebyshev polynomials)
Thanks.

Hint: square it, then make use of the fact that the 5 fifth roots of unity sum to 0.
