# 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