Consider the equation expressed the fifth root of unity: $z^5-1=0$
To show that: $$2(\cos(\frac{2\pi}{5})+\cos(\frac{4\pi}{5}))=-1\\4\cos(\frac{2\pi}{5})\cos(\frac{4\pi}{5})=-1$$
I have already shown the first one by using the sum of the root is zero and the truth that $\cos(\frac{2\pi}{5})=\cos(\frac{8\pi}{5})$ and $\cos(\frac{4\pi}{5})=\cos(\frac{6\pi}{5})$.
Now I am stacking on the second one and totally have no idea about how to do it.
Hint: The sum of the five roots is zero because $z^5-1$ has no fourth-degree term. What does the fact that there is no third-degree term tell you?