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
\(\ds \cosh 2 x\) | \(=\) | \(\ds \map \cosh {x + x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cosh x \cosh x + \sinh x \sinh x\) | 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