Tangent of Sum

Theorem

$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$

where $\tan$ is tangent.

$\map \tan {a - b} = \dfrac {\tan a - \tan b} {1 + \tan a \tan b}$

$\ds \map \tan {a + b}$

$=$

$\ds \frac {\map \sin {a + b} } {\map \cos {a + b} }$

$\ds $

$=$

$\ds \frac {\sin a \cos b + \cos a \sin b} {\cos a \cos b - \sin a \sin b}$

$\ds $

$=$

$\ds \frac {\frac {\sin a} {\cos a} + \frac {\sin b} {\cos b} } {1 - \frac {\sin a \sin b} {\cos a \cos b} }$

dividing top and bottom by $\cos a \cos b$

$\ds $

$=$

$\ds \frac {\tan a + \tan b} {1 - \tan a \tan b}$

$\blacksquare$

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(17)$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.36$
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): addition formulae
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): addition formulae
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): compound angle formulae (in trigonometry)
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Addition formulae
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Addition formulae