Mathematics reference
Hyperbolic trigonometry identities
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Various identities essential in hyperbolic trigonometry.

Introduction.
  • x and y are independent variables,
  • e is the base of the natural logarithm.
  • d is the differential operator,
  • int is the integral symbol,
  • C is the constant of integration.
Identities.
sinh x = (ex - e-x)/2 equation 1
cosh x = (ex + e-x)/2 equation 2
sech x = 1/cosh x equation 3
csch x = 1/sinh x equation 4
tanh x = sinh x/cosh x equation 5
coth x = 1/tanh x equation 6
cosh2 x - sinh2 x = 1 equation 7
tanh2 x + sech2 x = 1 equation 8
coth2 x - csch2 x = 1 equation 9
sinh (x + y) = sinh x cosh y + cosh x sinh y equation 10
sinh (x - y) = sinh x cosh y - cosh x sinh y equation 11
cosh (x + y) = cosh x cosh y + sinh x sinh y equation 12
cosh (x - y) = cosh x cosh y - sinh x sinh y equation 13
sinh2 x = [-1 + cosh (2 x)]/2 equation 14
sinh (2 x) = 2 sinh x cosh x equation 15
cosh2 x = [1 + cosh (2 x)]/2 equation 16
cosh (2 x) = cosh2 x + sinh2 x equation 17
arcsinh x = ln [x + (x2 + 1)1/2] equation 18
arccosh x = ln [x + (x2 - 1)1/2] equation 19
arctanh x = (1/2) ln [(1 + x)/(1 - x)] equation 20
arccoth x = (1/2) ln [(x + 1)/(x - 1)] equation 21
arcsech x = ln [[1 + (1 - x2)1/2]/x] equation 22
arccsch x = ln [[1 + (1 + x2)1/2]/|x|] equation 23
tanh (2 x) = 2 tanh x/(1 + tanh2 x) equation 24
coth x - tanh x = 2 csch (2 x) equation 25
coth x + tanh x = 2 coth (2 x) equation 26
(d/dx) sinh x = cosh x equation 27
(d/dx) cosh x = sinh x equation 28
(d/dx) tanh x = sech2 x equation 29
(d/dx) coth x = -csch2 x equation 30
(d/dx) sech x = -sech x tanh x equation 31
(d/dx) csch x = -csch x coth x equation 32
(d/dx) arcsinh x = 1/(x2 + 1)1/2 equation 33
(d/dx) arccosh x = 1/(x2 - 1)1/2 equation 34
(d/dx) arctanh x = 1/(1 - x2) equation 35
(d/dx) arccoth x = 1/(1 - x2) equation 36
(d/dx) arcsech x = -1/[x (1 - x2)1/2] equation 37
(d/dx) arccsch x = -1/[|x| (1 + x2)1/2] equation 38
Integrals.
int cosh x dx = sinh x + C equation 39
int sinh x dx = cosh x + C equation 40
int sech2 x dx = tanh x + C equation 41
int csch2 x dx = -coth x + C equation 42
int sech x tanh x dx = -sech x + C equation 43
int csch x coth x dx = -csch x + C equation 44
int dx/(x2 + 1)1/2 = arcsinh x + C equation 45
int dx/(x2 - 1)1/2 = arccosh x + C equation 46
int dx/(1 - x2) = arctanh x + C equation 47
int dx/(1 - x2) = arccoth x + C equation 48
int dx/[x (1 - x2)1/2] = -arcsech x + C equation 49
int dx/[|x| (1 + x2)1/2] = -arccsch x + C equation 50
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