Triple Angle Formulas/Cosine/Proof 3

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

$\blacksquare$


Sources