Hyperbolic Tangent of Complex Number/Formulation 4

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

Let $i$ be the imaginary unit.

Then:
 * $\tanh \paren {a + b i} = \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \cos 2 b}$

where:
 * $\tanh$ denotes the hyperbolic tangent function
 * $\sin$ denotes the real sine function
 * $\cos$ denotes the real cosine function
 * $\sinh$ denotes the hyperbolic sine function
 * $\cosh$ denotes the hyperbolic cosine function.

Also see

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