The hyperbolic cosine function is defined as

or generally,

.
Look at the graph below, try to slide the value of 'a' to different values to view the changes on graph.

Now, take a look at the graph of the reciprocal of hyperbolic cosine function, i.e. hyperbolic secant function, which is defined as

Observe that when a = 1,

- as , , thus ;
- as , , thus ;
- as , , thus .

Again, slide the value of 'a' to view the changes of the graph.

Betty, Created with GeoGebra

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