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.

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