Hyperbolic Secant of Complex Number

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

Let $i$ be the imaginary unit.

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

where:
 * $\sech$ denotes the hyperbolic secant 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 Tangent of Complex Number
 * Hyperbolic Cosecant of Complex Number
 * Hyperbolic Cotangent of Complex Number