Equivalence of Definitions of Real Hyperbolic Tangent

Theorem

The following definitions of the concept of Real Hyperbolic Tangent are equivalent:

Definition 1

The real hyperbolic tangent function is defined on the real numbers as:

$\tanh: \R \to \R$:

$\forall x \in \R: \tanh x := \dfrac {e^z - e^{-x} } {e^z + e^{-x} }$

Definition 2

The real hyperbolic tangent function is defined on the real numbers as:

$\tanh: \R \to \R$:

$\forall x \in \R: \tanh x := \dfrac {\sinh x} {\cosh x}$

where:

$\sinh$ is the real hyperbolic sine

$\cosh$ is the real hyperbolic cosine

Proof

\(\ds \tanh x\)

\(=\)

\(\ds \dfrac {\sinh x} {\cosh x}\)

Definition 2 of Real Hyperbolic Tangent

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {\dfrac {e^x - e^{-x} } 2} } {\paren {\dfrac {e^x + e^{-x} } 2} }\)

Definition of Real Hyperbolic Sine, Definition of Real Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \dfrac {e^x - e^{-x} } {e^x + e^{-x} }\)

simplification

\(\ds \)

\(=\)

\(\ds \tanh x\)

Definition 1 of Real Hyperbolic Tangent

$\blacksquare$