Tangent of Sum

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Theorem

$\displaystyle \tan \left({a + b}\right) = \frac {\tan a + \tan b} {1 - \tan a \tan b}$

where $\tan$ is tangent.

$\displaystyle \tan \left({a - b}\right) = \frac {\tan a - \tan b} {1 + \tan a \tan b}$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \tan \left({a + b}\right)$	$=$	$\displaystyle \frac {\sin \left({a + b}\right)} {\cos \left({a + b}\right)}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of tangent
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\sin a \cos b + \cos a \sin b} {\cos a \cos b - \sin a \sin b}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sine and Cosine of Sum
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\frac {\sin a} {\cos a} + \frac{\sin b} {\cos b} } {1 + \frac {\sin a \sin b} {\cos a \cos b} }$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by dividing the numerator and denominator by $\cos a \cos b$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\tan a + \tan b} {1 - \tan a \tan b}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of tangent

$\blacksquare$

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \tan \left({a - b}\right)$	$=$	$\displaystyle \frac {\tan a + \tan \left({-b}\right)} {1 - \tan a \tan \left({-b}\right)}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Main result
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\tan a - \tan b} {1 + \tan a \tan b}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Tangent Function is Odd

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \tan \left({a + b}\right)\)	\(=\)	\(\displaystyle \frac {\sin \left({a + b}\right)} {\cos \left({a + b}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of tangent
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\sin a \cos b + \cos a \sin b} {\cos a \cos b - \sin a \sin b}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Sum
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\frac {\sin a} {\cos a} + \frac{\sin b} {\cos b} } {1 + \frac {\sin a \sin b} {\cos a \cos b} }\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by dividing the numerator and denominator by $\cos a \cos b$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\tan a + \tan b} {1 - \tan a \tan b}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of tangent

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \tan \left({a - b}\right)\)	\(=\)	\(\displaystyle \frac {\tan a + \tan \left({-b}\right)} {1 - \tan a \tan \left({-b}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Main result
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\tan a - \tan b} {1 + \tan a \tan b}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Tangent Function is Odd