# Double Angle Formulas/Hyperbolic Cosine

## Theorem

$\cosh 2 x = \cosh^2 x + \sinh^2 x$

where $\cosh$ and $\sinh$ denote hyperbolic cosine and hyperbolic sine respectively.

### Corollary 1

$\cosh 2 x = 2 \cosh^2 x - 1$

### Corollary 2

$\cosh 2 x = 1 + 2 \sinh^2 x$

### Corollary 3

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

## Proof

 $\displaystyle \cosh 2 x$ $=$ $\displaystyle \map \cosh {x + x}$ $\displaystyle$ $=$ $\displaystyle \cosh x \cosh x + \sinh x \sinh x$ Hyperbolic Cosine of Sum $\displaystyle$ $=$ $\displaystyle \cosh^2 x + \sinh^2 x$

$\blacksquare$