Higher Engineering Mathematics, Sixth Edition

Hyperbolic functions 45

5.3 Hyperbolic identities

For every trigonometric identity there is a corres-
pondinghyperbolicidentity.Hyperbolicidentitiesmay
be proved by either

(i) replacing shx by

ex−e−x

2

and chx by

ex+e−x

2

,or

(ii) by usingOsborne’s rule, which states:‘the six

trigonometric ratios used in trigonometrical iden-

tities relating general angles may be replaced by

their corresponding hyperbolic functions, but the

sign of any direct or implied product of two sines

must be changed’.

For example, since cos^2 x+sin^2 x=1 then, by
Osborne’s rule, ch^2 x−sh^2 x=1, i.e. the trigonomet-
ric functions have been changed to their corresponding
hyperbolicfunctions and since sin^2 xis a product of two
sines the sign is changed from+to−. Table 5.1 shows
some trigonometric identities and their corresponding
hyperbolic identities.

Problem 9. Prove the hyperbolic identities

(a) ch^2 x−sh^2 x=1(b)1−th^2 x=sech^2 x

(c) coth^2 x− 1 =cosech^2 x.

(a) chx+shx=

(

ex+e−x

2

)

+

(

ex−e−x

2

)

=ex

chx−shx=

(

ex+e−x

2

)

−

(

ex−e−x

2

)

=e+−x

(chx+shx)(chx−shx)=(ex)(e−x)=e^0 = 1

i.e.ch^2 x−sh^2 x= 1 (1)

(b) Dividing each term in equation (1) by ch^2 x

gives:

ch^2 x

ch^2 x

−

sh^2 x

ch^2 x

=

1

ch^2 x

i.e. 1 −th^2 x=sech^2 x

Table 5.1

Trigonometric identity Corresponding hyperbolic identity

cos^2 x+sin^2 x= 1 ch^2 x−sh^2 x= 1

1 +tan^2 x=sec^2 x 1 −th^2 x=sech^2 x

cot^2 x+ 1 =cosec^2 x coth^2 x− 1 =cosech^2 x

Compound angle formulae

sin(A±B)=sinAcosB±cosAsinB sh(A±B)=shAchB±chAshB

cos(A±B)=cosAcosB∓sinAsinB ch(A±B)=chAchB±shAshB

tan(A±B)=

tanA±tanB

1 ∓tanAtanB

th(A±B)=

thA±thB

1 ±thAthB

Double angles

sin2x=2sinxcosx sh2x=2shxchx

cos2x=cos^2 x−sin^2 x ch2x=ch^2 x+sh^2 x

=2cos^2 x− 1 =2ch^2 x− 1

= 1 −2sin^2 x = 1 +2sh^2 x

tan2x=

2tanx

1 −tan^2 x

th2x=

2thx

1 +th^2 x