Quintuple Angle Formulas/Cosine/Proof 1

Theorem

$\cos 5 \theta = 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$

$\ds \cos 5 \theta$

$=$

$\ds \map \cos {4 \theta + \theta}$

$\ds $

$=$

$\ds \cos 4 \theta \cos \theta - \sin 4 \theta \sin \theta$

$\ds $

$=$

$\ds \paren {8 \cos^4 \theta - 8 \cos^2 \theta + 1} \cos \theta - \paren {4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta} \sin \theta$

$\ds $

$=$

$\ds 8 \cos^5 \theta - 8 \cos^3 \theta + \cos \theta - 4 \sin^2 \theta \cos \theta - 8 \sin^4 \theta \cos \theta$

multiplying out

$\ds $

$=$

$\ds 8 \cos^5 \theta - 8 \cos^3 \theta + \cos \theta - 4 \paren {1 - \cos^2 \theta} \cos \theta - 8 \paren {1 - \cos^2 \theta}^2 \cos \theta$

$\ds $

$=$

$\ds 16 \cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$

multiplying out and gathering terms

$\blacksquare$