# Tangent Half-Angle Substitution for Cosine

Jump to navigation Jump to search

## 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.

## Proof

 $\displaystyle \cos 2 \theta$ $=$ $\displaystyle \cos^2 \theta - \sin^2 \theta$ Double Angle Formula for Cosine $\displaystyle$ $=$ $\displaystyle \paren {\cos^2 \theta - \sin^2 \theta} \frac {\cos^2 \theta}{\cos^2 \theta}$ $\displaystyle$ $=$ $\displaystyle \paren {1 - \tan^2 \theta} \cos^2 \theta$ Tangent is Sine divided by Cosine $\displaystyle$ $=$ $\displaystyle \frac {1 - \tan^2 \theta} {\sec^2 \theta}$ Secant is Reciprocal of Cosine $\displaystyle$ $=$ $\displaystyle \frac {1 - \tan^2 \theta} {1 + \tan^2 \theta}$ Difference of Squares of Secant and Tangent

$\blacksquare$