# Triple Angle Formulas/Cosine/Proof 1

$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$
 $\displaystyle \cos 3 \theta$ $=$ $\displaystyle \cos \paren {2 \theta + \theta}$ $\displaystyle$ $=$ $\displaystyle \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta$ 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
$\blacksquare$