Cotangent of Complex Number/Formulation 3

Theorem
Let $a$ and $b$ be real numbers.

Let $i$ be the imaginary unit.

Then:
 * $\cot \paren {a + b i} = \dfrac {- \cot a - \cot a \coth^2 b} {\cot^2 a + \coth^2 b} + \dfrac {\cot^2 a \coth b - \coth b} {\cot^2 a + \coth^2 b} i$

where:
 * $\cot$ denotes the cotangent function (real and complex)
 * $\coth$ denotes the hyperbolic cotangent function.

Also see

 * Sine of Complex Number
 * Cosine of Complex Number
 * Tangent of Complex Number
 * Cosecant of Complex Number
 * Secant of Complex Number