Question

Prove that cosπ/5 + cos3π/5 = 1/2

Answer 1

Notice that equation $x^5-1=0$ has five roots, whose sum equals $0$ (from Vieta's theorem). It also means that their real parts also sum up to zero, i.e. $\cos\dfrac{2\pi}{5}+\cos\dfrac{4\pi}{5}-\cos\dfrac{\pi}{5}-\cos\dfrac{3\pi}{5}+1 = 0 \Leftrightarrow \cos\dfrac{\pi}{5}+\cos\dfrac{3\pi}{5} = 1+\cos\dfrac{2\pi}{5}+\cos\dfrac{4\pi}{5}$

By trig identities we have $\cos\dfrac{2\pi}{5}+\cos\dfrac{3\pi}{5}+\cos\dfrac{4\pi}{5} + \cos\dfrac{\pi}{5} = 2\cos\dfrac{\pi}{2}\cdot\ldots + 2\cos\dfrac{\pi}{2}\cdot\ldots = 0$ (the same result can be obtained by symmetry of the picture), so  $\cos\dfrac{\pi}{5}+\cos\dfrac{3\pi}{5} = 1+\cos\dfrac{2\pi}{5}+\cos\dfrac{4\pi}{5} = 1-\left(\cos\dfrac{\pi}{5}+\cos\dfrac{3\pi}{5} \right) \Leftrightarrow \cos\dfrac{\pi}{5}+\cos\dfrac{3\pi}{5} =1/2$.