# Powers of Cosine or Sine by Complex Number

Show that $\cos 4 \theta = 8 \cos^4 \theta-8 \cos^2 \theta + 1$.

(a)  By considering the real part of $z^4$, prove $\cos 4\theta = \cos^4 \theta-6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta$ by letting $z = \cos \theta + i \sin \theta$.

\require{AMSsymbols} \begin{align} \displaystyle \require{color} z &= \cos \theta + i \sin \theta \\ z^4 &= (\cos \theta + i \sin \theta)^4 \\ & = \cos 4\theta + i \sin 4\theta \cdots (1) &\color{red} \text{by De Moivre’s theorem}\\ z^4 &= (\cos \theta + i \sin \theta)^4 \\ &= \cos^4 \theta + 4 i \cos^3\theta \sin \theta + 6 i^2 \cos^2 \theta \sin ^2 \theta + 4i \cos \theta \sin^3 \theta + i^4 \sin \theta \\ &= \cos^4 \theta + 4 i \cos^3\theta \sin \theta-6 \cos^2 \theta \sin ^2 \theta + 4i \cos \theta \sin^3 \theta + \sin^4 \theta \\ &= (\cos^4 \theta-6 \cos^2 \theta \sin ^2 \theta + \sin^4 \theta ) + i(4 \cos^3\theta \sin \theta + 4 \cos \theta \sin^3 \theta) \cdots (2) \\ \end{align}
Equate only the real component of the expansion of $(1)$ and $(2)$.
$\therefore \cos 4\theta = \cos^4 \theta-6 \cos^2 \theta \sin ^2 \theta + \sin^4 \theta$

(b)   Hence, find an expression for $\sin 4\theta$ involving powers of $\sin \theta$ and $\cos \theta$.

\require{AMSsymbols} \begin{align} \require{color} \sin 4\theta &= 4 \cos^3\theta \sin \theta + 4 \cos \theta \sin^3 \theta &\color{red} \text{equating the imaginery parts}\\ \end{align}

(c)   Hence, find an expression for $\cos 4\theta$ involving only powers of $\cos \theta$.

\require{AMSsymbols} \begin{align} \require{color} \cos 4\theta &= \cos^4 \theta-6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta &\color{red} \text{equating the real parts}\\ &= \cos^4 \theta-6 \cos^2 \theta (1-\cos^2 \theta) + (1-\cos^2 \theta)^2 \\ &= \cos^4 \theta-6 \cos^2 \theta + 6 \cos^4 \theta + 1-2 \cos^2 \theta + \cos^4 \theta \\ &= 8 \cos^4 \theta-8 \cos^2 \theta + 1 \end{align}

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