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

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Theorem

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


Proof

\(\ds \) \(\) \(\ds \paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5\)
\(\ds \) \(=\) \(\ds \frac {\paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5} {\paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5} \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5\)
\(\ds \) \(=\) \(\ds \paren {\sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5\) Complex Division Examples: $\dfrac {1 + \sin \theta + i \cos \theta} {1 + \sin \theta - i \cos \theta}$
\(\ds \) \(=\) \(\ds i^5 \paren {-i \sin \dfrac \pi 5 + \cos \dfrac \pi 5}^5 \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5\) multiplying left hand term by $i^5 \times -i = 1$
\(\ds \) \(=\) \(\ds i \paren {\cos \pi - i \sin \pi} \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5\) De Moivre's Formula
\(\ds \) \(=\) \(\ds i \paren {-1} \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5\) Sine of Straight Angle and Cosine of Straight Angle
\(\ds \) \(=\) \(\ds -i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5\)
\(\ds \) \(=\) \(\ds 0\)


Sources

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