Double Angle Formulas/Cosine/Proof 1
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Theorem
- $\cos 2 \theta = \cos^2 \theta - \sin^2 \theta$
Proof
\(\ds \cos 2 \theta + i \sin 2 \theta\) | \(=\) | \(\ds \paren {\cos \theta + i \sin \theta}^2\) | De Moivre's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos^2 \theta + i^2 \sin^2 \theta + 2 i \cos \theta \sin \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos^2 \theta - \sin^2 \theta + 2 i \cos \theta \sin \theta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos 2 \theta\) | \(=\) | \(\ds \cos^2 \theta - \sin^2 \theta\) | equating real parts |
$\blacksquare$