Triple Angle Formulas/Cosine/Proof 1

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Theorem

$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$


Proof

\(\displaystyle \cos 3 \theta\) \(=\) \(\displaystyle \cos \paren {2 \theta + \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta\) Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \paren {\cos^2 \theta - \sin^2 \theta} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta\) Double Angle Formula for Cosine and Double Angle Formula for Sine
\(\displaystyle \) \(=\) \(\displaystyle \cos^3 \theta - \sin^2 \theta \cos \theta - 2 \sin^2 \theta \cos \theta\)
\(\displaystyle \) \(=\) \(\displaystyle \cos^3 \theta - \paren {1 - \cos^2 \theta} \cos \theta - 2 \paren {1 - \cos^2 \theta} \cos \theta\) Sum of Squares of Sine and Cosine
\(\displaystyle \) \(=\) \(\displaystyle \cos^3 \theta - \cos \theta + \cos^3 \theta - 2 \cos \theta + 2 \cos^3 \theta\) multiplying out
\(\displaystyle \) \(=\) \(\displaystyle 4 \cos^3 \theta - 3 \cos \theta\) gathering terms

$\blacksquare$