
25.4. Hyperbolic Functions 1415
25.4.2 Simplest Relations
cosh
2
x − sinh
2
x = 1,
sinh(−x) = −sinh x,
tanh x =
sinh x
cosh x
,
tanh(−x) = −tanh x,
1 −tanh
2
x =
1
cosh
2
x
,
tanh x coth x = 1,
cosh(−x) = cosh x,
coth x =
cosh x
sinh x
,
coth(−x) = −coth x,
coth
2
x − 1 =
1
sinh
2
x
.
25.4.3 Relations between Hyperbolic Functions of Single Argument
(x ≥ 0)
sinh x =
p
cosh
2
x − 1 =
tanh x
p
1 −tanh
2
x
=
1
p
coth
2
x − 1
,
cosh x =
p
sinh
2
x + 1 =
1
p
1 −tanh
2
x
=
coth x
p
coth
2
x − 1
,
tanh x =
sinh x
p
sinh
2
x + 1
=
p
cosh
2
x −1
cosh x
=
1
coth x
,
coth x =
p
sinh
2
x + 1
sinh x
=
cosh x
p
cosh
2
x −1
=
1
tanh x
.
25.4.4 Addition and Subtraction of Hyperbolic Functions
sinh x + sinh y = 2 sinh
x + y
2
cosh
x − y
2
,
sinh x − sinh y = 2 sinh ...