COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
cothxcothxtanhx
x2
4
− 2
− 4
− 2 − 1 1 2
Figure 3.13 Graphs of tanhxand cothx.metric functions transparent. The similarity in their calculus is discussed further
in subsection 3.7.6.
3.7.3 Identities of hyperbolic functionsThe analogies between trigonometric functions and hyperbolic functions having
been established, we should not be surprised that all the trigonometric identities
also hold for hyperbolic functions, with the following modification. Wherever
sin^2 xoccurs it must be replaced by−sinh^2 x, and vice versa. Note that this
replacement is necessary even if the sin^2 xis hidden, e.g. tan^2 x=sin^2 x/cos^2 x
and so must be replaced by (−sinh^2 x/cosh^2 x)=−tanh^2 x.
Find the hyperbolic identity analogous tocos^2 x+sin^2 x=1.Using the rules stated above cos^2 xis replaced by cosh^2 x,andsin^2 xby−sinh^2 x,andso
the identity becomes
cosh^2 x−sinh^2 x=1.This can be verified by direct substitution, using the definitions of coshxand sinhx;see
(3.38) and (3.39).
Some other identities that can be proved in a similar way aresech^2 x=1−tanh^2 x, (3.48)
cosech^2 x=coth^2 x− 1 , (3.49)sinh 2x=2sinhxcoshx, (3.50)cosh 2x=cosh^2 x+ sinh^2 x. (3.51)