Hyperbolic Tangent Half-Angle Substitution for Sine

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Corollary to Double Angle Formula for Hyperbolic Sine

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

where $\sin$ and $\tan$ denote hyperbolic sine and hyperbolic tangent respectively.


Proof

\(\ds \sinh 2 x\) \(=\) \(\ds 2 \sinh x \cosh x\) Double Angle Formula for Hyperbolic Sine
\(\ds \) \(=\) \(\ds 2 \sinh x \cosh x \frac {\cosh x} {\cosh x}\)
\(\ds \) \(=\) \(\ds 2 \tanh x \cosh^2 x\) Definition 2 of Hyperbolic Tangent
\(\ds \) \(=\) \(\ds \frac {2 \tanh x} {\sech^2 x}\) Definition 2 of Hyperbolic Secant
\(\ds \) \(=\) \(\ds \frac {2 \tanh x} {1 - \tanh^2 x}\) Sum of Squares of Hyperbolic Secant and Tangent

$\blacksquare$