# Rewrite Expressions in Sine and Cosine involving Powers of Cosine or Sine by Complex Number

Let \( z = \cos \theta + i \sin \theta \). (a) By considering the real part of \( z^4 \), prove \( \cos 4\theta = \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta \). \( \begin{align} \displaystyle \require{color} z &= \cos \theta + i \sin \theta \\ z^4 &= (\cos [...]