Gudermannian function

The Gudermannian function relates the circular and hyperbolic trig functions without resorting to complex numbers. Given
${\rm {gd}}(x)=2\arctan e^{x}-{\pi \over 2}$
then

sinh=tan·gd	cosh=sec·gd	csch=cot·gd
tanh=sin·gd	sech=cos·gd	coth=csc·gd

The inverse, ${\rm {gd}}^{-1}(\phi )=\ln(\tan \phi +\sec \phi )$, also turns up as the core of the Mercator projection.

Ref: CRC Handbook of Mathematical Sciences 5th ed. pp323–5