Hyperbolic tangent

The hyperbolic tangent is sinh divided by cosh and approaches plus or minus one at the two infinities.

For $|x|<1$, the inverse hyperbolic tangent is

The Taylor series at zero beginsso $\tanh x\sim x$ as $x\to 0$.

The exponential definitions of the hyperbolic functions imply identities such as

The identity ${\cosh }^{2}x-{\sinh }^{2}x=1$ implies

The identity follows directly from the exponential definitions of the hyperbolic functions.