Sum of 1 + sin pi by 5 plus i cos pi by 5 to Fifth Power plus i times its Conjugate

Theorem

 * $\paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 = 0$