How do you prove cos 3 theta = 4 cos^3 theta - 3 cos theta?

Jun 6, 2016

Proof is given below.

Explanation:

$\cos 3 \theta = \cos \left(2 \theta + \theta\right)$

$= \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta$

$= \left({\cos}^{2} \theta - {\sin}^{2} \theta\right) \cos \theta - 2 \sin \theta \cos \theta \sin \theta$

$= {\cos}^{3} \theta - {\sin}^{2} \cos \theta - 2 {\sin}^{2} \theta \cos \theta$

$= \cos \theta \left({\cos}^{2} \theta - {\sin}^{2} \theta - 2 {\sin}^{2} \theta\right)$

$= \cos \theta \left({\cos}^{2} \theta - 3 {\sin}^{2} \theta\right)$

$= {\cos}^{3} \theta - 3 {\sin}^{2} \theta \cos \theta$

$= {\cos}^{3} \theta - 3 \left(1 - {\cos}^{2} \theta\right) \cos \theta$

$= {\cos}^{3} \theta - 3 \cos \theta + 3 {\cos}^{3} \theta$

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