Double Angle Formulas/Tangent/Proof 1

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Theorem

$\tan 2 \theta = \dfrac {2 \tan \theta} {1 - \tan^2 \theta}$


Proof

\(\ds \tan 2 \theta\) \(=\) \(\ds \frac {\sin 2 \theta} {\cos 2 \theta}\) Tangent is Sine divided by Cosine
\(\ds \) \(=\) \(\ds \frac {2 \cos \theta \sin \theta} {\cos^2 \theta - \sin^2 \theta}\) Double Angle Formula for Sine and Double Angle Formula for Cosine
\(\ds \) \(=\) \(\ds \frac {\frac {2 \cos \theta \sin \theta} {\cos^2 \theta} } {\frac {\cos^2 \theta - \sin^2 \theta} {\cos^2 \theta} }\) dividing numerator and denominator by $\cos^2 \theta$
\(\ds \) \(=\) \(\ds \frac {2 \tan \theta} {1 - \tan^2 \theta}\) simplifying: Tangent is Sine divided by Cosine

$\blacksquare$