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

\(\displaystyle \cosh 2 x\) \(=\) \(\displaystyle \cosh^2 x + \sinh^2 x\) Double Angle Formula for Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \left({\cosh^2 x + \sinh^2 x}\right) \frac {\cosh^2 x}{\cosh^2 x}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({1 + \tanh^2 x}\right) \cosh^2 x\) Definition of Hyperbolic Tangent
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 + \tanh^2 x} {\operatorname{sech}^2 x}\) Definition of Hyperbolic Secant
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 + \tanh^2 x} {1 - \tanh^2 x}\) Sum of Squares of Hyperbolic Secant and Tangent

$\blacksquare$