2 Hyperbolic identities

The hyperbolic functions cosh x , sinh x satisfy similar (but not exactly equivalent) identities to those satisfied by cos x , sin x . We note first some basic notation similar to that employed with trigonometric functions:

cosh n x means ( cosh x ) n sinh n x means ( sinh x ) n n 1

In the special case that n = 1 we do not use cosh 1 x and sinh 1 x to mean 1 cosh x and 1 sinh x respectively. The notation cosh 1 x and sinh 1 x is reserved for the inverse functions of cosh x and sinh x respectively.

Task!

Show that cosh 2 x sinh 2 x 1 for all x .

  1. First, express cosh 2 x in terms of the exponential functions e x , e x :

    1 4 ( e x + e x ) 2 1 4 [ ( e x ) 2 + 2 e x e x + ( e x ) 2 ] 1 4 [ e 2 x + 2 e x x + e 2 x ] 1 4 [ e 2 x + 2 + e 2 x ]

  2. Similarly, express sinh 2 x in terms of e x and e x :

    1 4 ( e x e x ) 2 1 4 [ ( e x ) 2 2 e x e x + ( e x ) 2 ] 1 4 [ e 2 x 2 e x x + e 2 x ] 1 4 [ e 2 x 2 + e 2 x ]

  3. Finally determine cosh 2 x sinh 2 x using the results from (1) and (2):

    cosh 2 x sinh 2 x 1 4 [ e 2 x + 2 + e 2 x ] 1 4 [ e 2 x 2 + e 2 x ] 1

As an alternative to the calculation in this Task we could, instead, use the relations

e x cosh x + sinh x e x cosh x sinh x

and remembering the algebraic identity ( a + b ) ( a b ) a 2 b 2 , we see that

( cosh x + sinh x ) ( cosh x sinh x ) e x e x 1 that is cosh 2 x sinh 2 x 1

Key Point 4

The fundamental identity relating hyperbolic functions is:

cosh 2 x sinh 2 x 1
This is the hyperbolic function equivalent of the trigonometric identity: cos 2 x + sin 2 x 1
Task!

Show that cosh ( x + y ) cosh x cosh y + sinh x sinh y .

First, express cosh x cosh y in terms of exponentials:

e x + e x 2 e y + e y 2 1 4 [ e x e y + e x e y + e x e y + e x e y ] 1 4 ( e x + y + e x + y + e x y + e x y )

Now express sinh x sinh y in terms of exponentials:

e x e x 2 e y e y 2 1 4 ( e x + y e x + y e x y + e x y )

Now express cosh x cosh y + sinh x sinh y in terms of a hyperbolic function:

cosh x cosh y + sinh x sinh y 1 2 ( e x + y + e ( x + y ) ) which we recognise as cosh ( x + y )

Other hyperbolic function identities can be found in a similar way. The most commonly used are listed in the following Key Point.

Key Point 5

Hyperbolic Identities

cosh 2 sinh 2 1

cosh ( x + y ) cosh x cosh y + sinh x sinh y

sinh ( x + y ) sinh x cosh y + cosh x sinh y

sinh 2 x 2 sinh x cosh y

cosh 2 x cosh 2 x + sinh 2 x or cosh 2 x 2 cosh 2 1 or cosh 2 x 1 + 2 sinh 2 x