How to derive the formulas for hyperbolic cotangent?

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

Hyperbolic cotangent, denoted as coth(x), is defined as the ratio of hyperbolic cosine and hyperbolic sine functions:

coth(x) = cosh(x) / sinh(x)

Using the definitions of hyperbolic cosine and sine functions:

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

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

Substituting these definitions into the formula for coth(x):

coth(x) = (e^x + e^(-x)) / (e^x - e^(-x))

To simplify this expression, we can multiply the numerator and denominator by e^x:

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

This is the formula for hyperbolic cotangent. It can also be expressed in terms of exponential functions:

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