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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)

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