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 |
$\blacksquare$