# Double Angle Formulas/Sine/Corollary

## Corollary to Double Angle Formula for Sine

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

where $\sin$ and $\tan$ denote sine and tangent respectively.

## Proof

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

$\blacksquare$