Sum of Cosines of 2 k pi over 5

Proof
Let $z_1, z_2, \dotsc, z_5$ be the complex $5$th roots of unity.

From Sum of Powers of Primitive Complex Roots of Unity, setting $s = 1$:


 * $z_1 + z_2 + z_3 + z_4 + z_5 = 0$

Thus from Complex 5th Roots of Unity:
 * $1 + \cis 72 \degrees + \cis 144 \degrees + \cis 216 \degrees + \cis 288 \degrees = 0$

Equating the real parts:
 * $1 + \cos 72 \degrees + \cos 144 \degrees + \cos 216 \degrees + \cos 288 \degrees = 0$


 * Sum of Cosines of 2 k pi over 5.png