MATH 018 DIFFERENTIAL CALCULUS
The Hyperbolic Identities
Prepared by: Dr. Jeffrey G. Capacite
ex − e− x
I. sinh(x) =
2
x
e + e− x
II. cosh(x) =
2
sinh(x) e x − e − x
III. tanh(x) =
=
cosh(x) e x + e − x
IV. sech(x) =
1
2
= x
cosh(x) e + e − x
V. csch(x) =
1
2
= x −x
sinh(x) e − e
cosh(x) e x + e − x
=
VI. coth(x) =
sinh(x) e x − e − x
VII. cosh2 (x) − si nh2 (x) = 1
VIII. 1 − tanh2 (x) = sech2 (x)
IX. coth2 (x) − 1 = csch2 (x)
X. sinh(x  y) = sinh(x)cosh(y)  cosh(x)sinh(y)
XI. cosh(x  y) = cosh(x)cosh(y)  sinh(x)sinh(y)
XII. tanh(x  y) =
tanh(x)  tanh(y)
1  tanh(x) tanh(y)
XIII. sinh(2x) = 2sinh(x)cosh(x)
XIV. cosh(2x) = cosh2 (x) + si nh2 (x)
XV. cosh2 (x) =
cosh(2x) + 1
2
XVI. sinh2 (x) =
cosh(2x) − 1
2
1