Cotangent of Sum

Theorem

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

where $\cot $ is cotangent.

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

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

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

dividing numerator and denominator by $\sin a \sin b$

$\ds $

$=$

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

$\blacksquare$