Hyperbolic identities
cosh² − sinh² = 1; double-angle and addition formulae.
Hyperbolic identities mirror trig with sign changes (Osborn's rule: replace sin² with −sinh² in trig identities).
Worked examples
cosh²x − sinh²x = 1.
sinh 2x = 2 sinh x cosh x.
cosh 2x = cosh²x + sinh²x = 1 + 2 sinh²x.
Frequently asked questions
Osborn's rule?
Take trig identity, replace sin²/cos² with −sinh²/cosh² etc. Useful mnemonic.
Why no negative signs in cosh 2x?
Both terms positive. Reflects the +1 in cosh² − sinh².