Tangent Half-Angle Substitution for Cosine

Corollary to Double Angle Formula for Cosine

$\cos 2 \theta = \dfrac {1 - \tan^2 \theta} {1 + \tan^2 \theta}$

where $\cos$ and $\tan$ denote cosine and tangent respectively.

$\ds \cos 2 \theta$

$=$

$\ds \cos^2 \theta - \sin^2 \theta$

$\ds $

$=$

$\ds \paren {\cos^2 \theta - \sin^2 \theta} \frac {\cos^2 \theta}{\cos^2 \theta}$

$\ds $

$=$

$\ds \paren {1 - \tan^2 \theta} \cos^2 \theta$

$\ds $

$=$

$\ds \frac {1 - \tan^2 \theta} {\sec^2 \theta}$

$\ds $

$=$

$\ds \frac {1 - \tan^2 \theta} {1 + \tan^2 \theta}$

$\blacksquare$