How to derive the identity for hyperbolic tangent?

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

Hyperbolic functions are defined in terms of exponential functions as follows:

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

To derive the identity for hyperbolic tangent, we start with the definition of hyperbolic tangent:

tanh(x) = sinh(x)/cosh(x)

We can then substitute the definitions of sinh(x) and cosh(x) in terms of exponential functions:

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

We can simplify this expression by multiplying the numerator and denominator by 2/e^x:

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

This is the identity for hyperbolic tangent. We can verify this identity by substituting a value for x and evaluating both sides of the equation. For example, if we let x = 1, we get:

tanh(1) = (e^1 - e^-1) / (e^1 + e^-1) = 0.7616...
(1 - e^-2) / (1 + e^-2) = 0.7616...

Both sides of the equation evaluate to the same value, so the identity is verified.