Cosecant of Complex Number

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

Let $i$ be the imaginary unit.

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

where:
 * $\csc$ denotes the complex cosecant 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
 * Tangent of Complex Number
 * Secant of Complex Number
 * Cotangent of Complex Number