the last of which is similar to the Pythagorean trigonometric identity.
It can be proved by comparing term by term the Taylor series of the two functions.
The following series are followed by a description of a subset of their domain of convergence, where the series is convergent and its sum equals the function.
The Gudermannian function gives a direct relationship between the circular functions, and the hyperbolic ones that does not involve complex numbers.
The decomposition of the exponential function in its even and odd parts gives the identities
Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers: