Double Angle Formulas/Tangent
Theorem
- $\tan 2 \theta = \dfrac {2 \tan \theta} {1 - \tan^2 \theta}$
where $\tan$ denotes tangent.
Corollary
Let $u = \tan \dfrac \theta 2$.
Then:
- $\tan \theta = \dfrac {2 u} {1 - u^2}$
Proof 1
| \(\ds \tan 2 \theta\) | \(=\) | \(\ds \frac {\sin 2 \theta} {\cos 2 \theta}\) | 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 |
$\blacksquare$
Proof 2
| \(\ds \tan 2 \theta\) | \(=\) | \(\ds \map \tan {\theta + \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\tan \theta + \tan \theta} {1 - \tan \theta \tan \theta}\) | Tangent of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 \tan \theta} {1 - \tan^2 \theta}\) |
$\blacksquare$
Proof 3
| \(\ds \frac {2 \tan \theta} {1 - \tan^2 \theta}\) | \(=\) | \(\ds \dfrac {2 i \dfrac {1 - e^{2 i \theta} } {1 + e^{2 i \theta} } } {1 - \paren {i \dfrac {1 - e^{2 i \theta} } {1 + e^{2 i \theta} } }^2}\) | Euler's Tangent Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 i \paren {1 - e^{2 i \theta} } \paren {1 + e^{2 i \theta} } } {\paren {1 + e^{2 i \theta} }^2 + \paren {1 - e^{2 i \theta} }^2}\) | multiplying both numerator and denominator by $\paren {1 + e^{2 i \theta} }^2$; $i$ is the imaginary unit | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 i \paren {1 - e^{4 i \theta} } } {1 + 2 e^{2 i \theta} + e^{4 i \theta} + 1 - 2 e^{2 i \theta} + e^{4 i \theta} }\) | Difference of Two Squares, Square of Sum, Square of Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 i \paren {1 - e^{4 i \theta} } } {2 + 2 e^{4 i \theta} }\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds i \dfrac {1 - e^{4 i \theta} } {1 + e^{4 i \theta} }\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \tan 2 \theta\) | Euler's Tangent Identity |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(19)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.40$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): double-angle formulae
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): double-angle formulae
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): double-angle formula (in trigonometry)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Double-angle formulae
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Double-angle formulae