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Hyperbolic Functions

Angle Sum and Difference Identities

sinh(x ± y) = sinh x cosh y ± cosh x sinh y

cosh(x ± y) = cosh x cosh y ± sinh x sinh y
tanh x ± tanh y sinh 2x ± sinh 2y
tanh(x ± y) = =
1 ± tanh x tanh y cosh 2x ± cosh 2y
1 ± coth x coth y sinh 2x ∓ sinh 2y
coth(x ± y) = =
coth x ± coth y cosh 2x − cosh 2y

Function Sum and Difference Identities

1 1
sinh x ± sinh y = 2 sinh (x ± y) cosh (x ∓ y)
2 2
1 1
cosh x + cosh y = 2 cosh (x + y) cosh (x − y)
2 2
1 1
cosh x − cosh y = 2 sinh (x + y) sinh (x − y)
2 2
sinh(x ± y)
tanh x ± tanh y =
cosh x cosh y
sinh(x ± y)
coth x ± coth y =
sinh x sinh y

Double Angle Identities

2
2
sinh 2x = 2 sinh x cosh x, cosh 2x = cosh x + sinh x

Half Angle Identities

x cosh x − 1 x cosh x + 1
sinh 2 = , cosh 2 =
2 2 2 2

Function Product Identities

1 1
sinh x sinh y = cosh(x + y) − cosh(x − y)
2 2
1 1
cosh x cosh y = cosh(x + y) + cosh(x − y)
2 2
1 1
sinh x cosh y = sinh(x + y) + sinh(x − y)
2 2

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