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

\(\ds \cosh 2 x\)

\(=\)

\(\ds \map \cosh {x + x}\)

\(\ds \)

\(=\)

\(\ds \cosh x \cosh x + \sinh x \sinh x\)

Hyperbolic Cosine of Sum

\(\ds \)

\(=\)

\(\ds \cosh^2 x + \sinh^2 x\)

$\blacksquare$

Sources

• 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