Double Angle Formulas/Hyperbolic Tangent/Proof 1

Theorem

$\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$

Proof

\(\ds \tanh 2 x\)

\(=\)

\(\ds \frac {\sinh 2 x} {\cosh 2 x}\)

Definition 2 of Hyperbolic Tangent

\(\ds \)

\(=\)

\(\ds \frac {2 \cosh x \sinh x} {\cosh^2 x + \sinh^2 x}\)

Double Angle Formula for Hyperbolic Sine and Double Angle Formula for Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \frac {\frac {2 \cosh x \sinh x} {\cosh^2 x} } {\frac {\cosh^2 x + \sinh^2 x} {\cosh^2 x} }\)

dividing top and bottom by $\cosh^2 x$

\(\ds \)

\(=\)

\(\ds \frac {2 \tanh x} {1 + \tanh^2 x}\)

Definition 2 of Hyperbolic Tangent

$\blacksquare$