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==Useful relations==
{{Anchor|Osborn}}
The hyperbolic functions satisfy many identities, all of them similar in form to the [[trigonometric identity|trigonometric identities]]. In fact, '''Osborn's rule'''<ref name="Osborn, 1902" /> states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for <math>\theta</math>, <math>2\theta</math>, <math>3\theta</math> or <math>\theta</math> and <math>\varphi</math> into a hyperbolic identity, by:
# expanding it completely in terms of integral powers of sines and cosines, # changing sine to sinh and cosine to cosh, and # switching the sign of every term containing a product of two sinhs. Odd and even functions:
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