Quadruple Angle Formulas/Cosine/Proof 1
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Theorem
- $\cos 4 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1$
Proof
| \(\ds \cos 4 \theta\) | \(=\) | \(\ds \cos \paren {2 \theta + 2 \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos 2 \theta \cos 2 \theta - \sin 2 \theta \sin 2 \theta\) | Cosine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\cos^2 \theta - \sin^2 \theta} \paren {\cos^2 \theta - \sin^2 \theta} - \paren {2 \sin \theta \cos \theta} \paren {2 \sin \theta \cos \theta}\) | Double Angle Formulas | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^4 \theta - 2 \cos^2 \theta \sin^2 \theta + \sin^4 \theta - 4 \cos^2 \theta \sin^2 \theta\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^4 \theta - 2 \cos^2 \theta \paren {1 - \cos^2 \theta} + \paren {1 - \cos^2 \theta}^2 - 4 \cos^2 \theta \paren {1 - \cos^2 \theta}\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 8 \cos^4 \theta - 8 \cos^2 \theta + 1\) | multiplying out and gathering terms |
$\blacksquare$