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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.

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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?

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  • $\begingroup$ Thanks, I got that. $\endgroup$ – Yuanming Luo Oct 17 '18 at 11:36

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