How to derive the formulas for hyperbolic secant?

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

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

sinh(x) = (e^x - e^-x)/2

Similarly, the hyperbolic cosine function is defined as:

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

The hyperbolic secant function is defined as the reciprocal of the hyperbolic cosine function:

sech(x) = 1/cosh(x)

Substituting the definition of cosh(x) into the formula for sech(x), we get:

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

Multiplying the numerator and denominator by 2e^x, we get:

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

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

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

This is the formula for hyperbolic secant.