Tangent Half-Angle Substitution for Sine
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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
\(\ds \sin 2 \theta\) | \(=\) | \(\ds 2 \sin \theta \cos \theta\) | Double Angle Formula for Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sin \theta \cos \theta \frac {\cos \theta} {\cos \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \tan \theta \cos^2 \theta\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \tan \theta} {\sec^2 \theta}\) | Secant is Reciprocal of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \tan \theta} {1 + \tan^2 \theta}\) | Difference of Squares of Secant and Tangent |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(26)$