# Hyperbolic Tangent Half-Angle Substitution for Cosine

$\cosh 2 x = \dfrac {1 + \tanh^2 x}{1 - \tanh^2 x}$
where $\cosh$ and $\tanh$ denote hyperbolic cosine and hyperbolic tangent respectively.
 $\displaystyle \cosh 2 x$ $=$ $\displaystyle \cosh^2 x + \sinh^2 x$ 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
$\blacksquare$