Double Angle Formulas/Hyperbolic Sine/Corollary

Corollary to Double Angle Formula for Hyperbolic Sine

$\map \sinh {2 \theta} = \dfrac {2 \tanh \theta} {1 - \tanh^2 \theta}$

where $\sinh$ and $\tanh$ denote hyperbolic sine and hyperbolic tangent respectively.

Proof

\(\ds \map \sinh {2 \theta}\)

\(=\)

\(\ds 2 \sinh \theta \cosh \theta\)

Double Angle Formula for Hyperbolic Sine

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds 2 \tanh \theta \cosh^2 \theta\)

Definition 2 of Hyperbolic Tangent

\(\ds \)

\(=\)

\(\ds \frac {2 \tanh \theta} {\sech^2 \theta}\)

Definition 2 of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \frac {2 \tanh \theta} {1 - \tanh^2 \theta}\)

Sum of Squares of Hyperbolic Secant and Tangent

$\blacksquare$