How to derive the identity for hyperbolic cotangent?

The identity for hyperbolic cotangent is derived using the definition of hyperbolic functions.

The hyperbolic cotangent function is defined as:

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

We can manipulate this expression using the definitions of hyperbolic sine and cosine:

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

Substituting these expressions into the definition of coth(x), we get:

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 identity for hyperbolic cotangent. We can use it to simplify expressions involving hyperbolic cotangent, or to prove other identities involving hyperbolic functions.

For example, we can use this identity to derive the identity for hyperbolic tangent:

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

Substituting the expressions for hyperbolic sine and cosine, we get:

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

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

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

This is the identity for hyperbolic tangent, which can also be used to simplify expressions involving hyperbolic tangent.