Tangent of Complex Number/Formulation 4

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

Let $i$ be the imaginary unit.

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

where:
 * $\tan$ denotes the complex 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

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