Double Angle Formulas/Sine/Corollary

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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

\(\displaystyle \sin 2 \theta\) \(=\) \(\displaystyle 2 \sin \theta \cos \theta\) 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

$\blacksquare$


Sources