Tangent of Complex Number/Formulation 3

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

Let $i$ be the imaginary unit.

Then:
 * $\tan \paren {a + b i} = \dfrac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \dfrac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i$

where:
 * $\tan$ denotes the tangent function (real and complex)
 * $\tanh$ denotes the hyperbolic tangent function.

Also see

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