Double Angle Formulas/Hyperbolic Cosine

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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

\(\displaystyle \cosh 2 x\) \(=\) \(\displaystyle \cosh \left({x + x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \cosh x \cosh x + \sinh x \sinh x\) Hyperbolic Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \cosh^2 x + \sinh^2 x\)

$\blacksquare$


Sources