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Hyperbolic Functions

Function Product Identities

1 1
sin x sin y = cos(x − y) − cos(x + y)
2 2
1 1
cos x cos y = cos(x − y) + cos(x + y)
2 2
1 1
sin x cos y = sin(x + y) + sin(x − y)
2 2
1 1
cos x sin y = sin(x + y) − sin(x − y)
2 2

Exponential Identities

ix
ix
e − e −ix e + e −ix
ix
e = cos x + i sin x, sin x = , cos x =
i2 2
G.2 Hyperbolic Functions

Exponential Identities

x
x
e − e −x e + e −x
sinh x = , cosh x =
2 2
x
sinh x e − e −x
tanh x = =
x
cosh x e + e −x
Reciprocal Identities

1 1 1
csch x = , sech x = , coth x =
sinh x cosh x tanh x

Pythagorean Identities

2
2
2
2
cosh x − sinh x = 1, tanh x + sech x = 1
Relation to Circular Functions

sinh(ix) = i sin x sinh x = −i sin(ix)
cosh(ix) = cos x cosh x = cos(ix)

tanh(ix) = i tan x tanh x = −i tan(ix)

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