Hyperbolic Tangent Half-Angle Substitution for Cosine
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Corollary to Double Angle Formula for Hyperbolic Cosine
- $\cosh 2 x = \dfrac {1 + \tanh^2 x}{1 - \tanh^2 x}$
where $\cosh$ and $\tanh$ denote hyperbolic cosine and hyperbolic tangent respectively.
Proof
| \(\ds \cosh 2 x\) | \(=\) | \(\ds \cosh^2 x + \sinh^2 x\) | Double Angle Formula for Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\cosh^2 x + \sinh^2 x} \frac {\cosh^2 x} {\cosh^2 x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {1 + \tanh^2 x} \cosh^2 x\) | Definition 2 of Hyperbolic Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 + \tanh^2 x} {\sech^2 x}\) | Definition 2 of Hyperbolic Secant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 + \tanh^2 x} {1 - \tanh^2 x}\) | Sum of Squares of Hyperbolic Secant and Tangent |
$\blacksquare$