addition and subtraction formulas for tangent

The addition formula for tangent will be achieved via brute from the addition formulas for sine and cosine.

$\begin{array}[]{rl}\tan(\alpha+\beta)&=\displaystyle\frac{\sin(\alpha+\beta)}{% \cos(\alpha+\beta)}\\ &\\ &=\displaystyle\frac{\sin\alpha\cos\beta+\cos\alpha\sin\beta}{\cos\alpha\cos% \beta-\sin\alpha\sin\beta}\\ &\\ &=\displaystyle\frac{\displaystyle\frac{\sin\alpha}{\cos\alpha}\cdot\frac{\cos% \beta}{\cos\beta}+\displaystyle\frac{\cos\alpha}{\cos\alpha}\cdot\frac{\sin% \beta}{\cos\beta}}{\displaystyle\frac{\cos\alpha}{\cos\alpha}\cdot\frac{\cos% \beta}{\cos\beta}-\frac{\sin\alpha}{\cos\alpha}\cdot\frac{\sin\beta}{\cos\beta% }}\\ &\\ &=\displaystyle\frac{\tan\alpha\cdot 1+1\cdot\tan\beta}{1\cdot 1-\tan\alpha% \tan\beta}\\ &\\ &=\displaystyle\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\end{array}$

Note that $\tan$ is an odd function, i.e. (http://planetmath.org/Ie) $\tan(-x)=-\tan x$ . This fact enables us to obtain the subtraction formula for tangent.

$\tan(\alpha-\beta)=\tan(\alpha+(-\beta))=\frac{\tan\alpha+\tan(-\beta)}{1-\tan% \alpha\tan(-\beta)}=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$

Title	addition and subtraction formulas for tangent
Canonical name	AdditionAndSubtractionFormulasForTangent
Date of creation	2013-03-22 16:59:04
Last modified on	2013-03-22 16:59:04
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	8
Author	Wkbj79 (1863)
Entry type	Derivation
Classification	msc 33B10
Classification	msc 26A09
Synonym	addition and subtraction formulae for tangent
Synonym	addition formula for tangent
Synonym	subtraction formula for tangent
Related topic	AdditionFormula
Related topic	DefinitionsInTrigonometry
Related topic	AngleBetweenTwoLines
Related topic	AdditionFormulas