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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)

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