Hyperbolic Identities

Hyperbolic functions satisfy identities that closely mirror their trigonometric counterparts, with predictable sign changes governed by Osborn's rule.

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1 The Fundamental Identities

cosh^2 x - sinh^2 x = 1

1 - tanh^2 x = sech^2 x

coth^2 x - 1 = cosech^2 x

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2 Osborn's Rule

To convert a trigonometric identity to its hyperbolic counterpart:

1. Replace every trig function with its hyperbolic analogue (sin -> sinh, cos -> cosh, etc.).
2. Whenever a product of two sinh terms appears, change the sign of that term.

Trigonometric	Hyperbolic
cos^2 + sin^2 = 1	cosh^2 - sinh^2 = 1
1 + tan^2 = sec^2	1 - tanh^2 = sech^2
sin(A+B) = sinA cosB + cosA sinB	sinh(A+B) = sinhA coshB + coshA sinhB
cos(A+B) = cosA cosB - sinA sinB	cosh(A+B) = coshA coshB + sinhA sinhB

Warning: Osborn's rule is a mnemonic, not a proof. In exams, you may need to prove identities from the exponential definitions.

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3 Addition Formulae

sinh(A + B) = sinh A cosh B + cosh A sinh B