The Hyperbolic Identities
Chapter Thirty
Syllabus topic Module 1, "1.1 Complex Numbers"
Pages 68 to 69 of 303
In one line
Every hyperbolic identity comes out of the exponential definitions in two lines, so none of them has to be remembered.
The fundamental one
Square both definitions and subtract.
cosh(x)^2 = (e^(2x) + 2 + e^(-2x))/4
sinh(x)^2 = (e^(2x) - 2 + e^(-2x))/4
cosh(x)^2 - sinh(x)^2 = 1
The two squares differ only in the middle term, so the subtraction leaves 4 over 4, which is one. That is the identity the whole family hangs on, and it is the hyperbolic counterpart of cos squared plus sin squared equals one, with a minus sign in place of the plus.
Divide it through by cosh squared, and then by sinh squared, for the other two forms.
1 - tanh(x)^2 = sech(x)^2
coth(x)^2 - 1 = csch(x)^2
The compound-angle formulae
Derived the same way, by multiplying out exponentials. Here they are, and every one was verified by machine.
sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y)
sinh(x - y) = sinh(x) cosh(y) - cosh(x) sinh(y)
cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y)
cosh(x - y) = cosh(x) cosh(y) - sinh(x) sinh(y)
Compare them with the circular ones. The sine formulae are identical in shape. The cosine formulae have their signs the other way round: the circular cos(x + y) has a minus and the hyperbolic cosh(x + y) has a plus.
And the tangent:
tanh(x + y) = (tanh(x) + tanh(y))/(1 + tanh(x) tanh(y))
where the circular version has a minus in the denominator.
The double-angle formulae
Put y equal to x in each of the above.
sinh(2x) = 2 sinh(x) cosh(x)
cosh(2x) = cosh(x)^2 + sinh(x)^2
tanh(2x) = 2 tanh(x)/(1 + tanh(x)^2)
And the three alternative forms of cosh 2x, obtained with the fundamental identity, which are the ones actually used in integration.
cosh(2x) = 2 cosh(x)^2 - 1
cosh(2x) = 1 + 2 sinh(x)^2
cosh(x)^2 = (cosh(2x) + 1)/2
sinh(x)^2 = (cosh(2x) - 1)/2
The last two are the hyperbolic versions of the formulae that let you integrate a squared cosine or sine, and they are needed for that purpose in the Laplace transform chapters.
The triple-angle formulae
sinh(3x) = 3 sinh(x) + 4 sinh(x)^3
cosh(3x) = 4 cosh(x)^3 - 3 cosh(x)
Compare with the circular ones from the De Moivre chapter: sin 3t is 3 sin t minus 4 sin cubed, and cos 3t is 4 cos cubed minus 3 cos. So the cosh formula is identical and the sinh one has a sign changed.
Sums to products
Useful in integration and occasionally asked.
sinh(x) + sinh(y) = 2 sinh((x + y)/2) cosh((x - y)/2)
sinh(x) - sinh(y) = 2 cosh((x + y)/2) sinh((x - y)/2)
cosh(x) + cosh(y) = 2 cosh((x + y)/2) cosh((x - y)/2)
cosh(x) - cosh(y) = 2 sinh((x + y)/2) sinh((x - y)/2)
The Hyperbolic Identities
How to answer "prove the identity" in an examination
Never by quoting another identity you also cannot prove. Always from the definitions, which is four lines at most.
Worked: prove that cosh(x + y) equals cosh x cosh y plus sinh x sinh y.
Write both products out in exponentials.
cosh(x) cosh(y) = (e^(x + y) + e^(x - y) + e^(-x + y) + e^(-x - y))/4
sinh(x) sinh(y) = (e^(x + y) - e^(x - y) - e^(-x + y) + e^(-x - y))/4
Add them. The two middle terms cancel in pairs and the outer terms double.
cosh(x) cosh(y) + sinh(x) sinh(y) = (e^(x + y) + e^(-x - y))/2
(e^(x + y) + e^(-(x + y)))/2 = cosh(x + y)
That is the result. Every identity on this page yields to exactly that procedure.
Check yourself
cosh(x)^2 - sinh(x)^2 = 1
sech(x)^2 + tanh(x)^2 = 1
sinh(2x) = 2 sinh(x) cosh(x)
cosh(2x) - 1 = 2 sinh(x)^2
sinh(x + y) sinh(x - y) = sinh(x)^2 - sinh(y)^2
tanh(x) + coth(x) = 2 coth(2x)
The last two are the kind of identity that looks hard and is not: write each side from the definitions, or from the compound-angle formulae above, and the two sides collapse onto each other.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself for the same subject.