Tangent Half-Angle Substitution for Sine

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.

$\ds \sin 2 \theta$

$=$

$\ds 2 \sin \theta \cos \theta$

$\ds $

$=$

$\ds 2 \sin \theta \cos \theta \frac {\cos \theta} {\cos \theta}$

$\ds $

$=$

$\ds 2 \tan \theta \cos^2 \theta$

$\ds $

$=$

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

$\ds $

$=$

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

$\blacksquare$

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(26)$