How to derive the identity for hyperbolic secant?

The identity for hyperbolic secant is derived from the definition of hyperbolic functions.

Hyperbolic functions are defined in terms of exponential functions. The hyperbolic secant function is defined as:

sech(x) = 1/cosh(x)

where cosh(x) is the hyperbolic cosine function.

To derive the identity for sech(x), we start with the definition of cosh(x):

cosh(x) = (e^x + e^(-x))/2

Substituting this into the definition of sech(x), we get:

sech(x) = 1/((e^x + e^(-x))/2)

Multiplying the numerator and denominator by 2, we get:

sech(x) = 2/(e^x + e^(-x))

Multiplying the numerator and denominator by e^x, we get:

sech(x) = 2e^x/(e^(2x) + 1)

This is the identity for hyperbolic secant. It can also be written as:

sech(x) = (e^x + e^(-x))/(e^(2x) + 1)

or

sech(x) = (1 - e^(-2x))/(e^(2x) + 1)