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.

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

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

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

$\ds \cosh 2 x$

$=$

$\ds \map \cosh {x + x}$

$\ds $

$=$

$\ds \cosh x \cosh x + \sinh x \sinh x$

$\ds $

$=$

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

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.25$: Double Angle Formulas
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cosh or ch
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hyperbolic function