# Using the fifth root of unity to show the cosine equation

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?