As you might suspect, there are also hyperbolic secant, co-secant and

cotangent functions. Let's investigate these hyperbolic trig

functions form the point of view of Taylor series co-secant.

For example, if we consider the hyperbolic cosine of X as one half E to

the X plus one half E to the minus X. Then it's clear how to compute the

Taylor's series. Instead of trying to take derivatives,

we'll simply substitute in the known series for e to the X, multiplied by one

half and then add to it, 1 1/2 times the series for e to the minus X.

Which, of course is the Taylor series for e to the X, with minus X substituted in.

Leaving negative signs at the odd powers of X.

By combining terms according to degree, we see that all of the odd degree terms

cancel. And we are left with only the even degree

terms. The same as in the expansion of E to the

X 1 plus X squared over 2 factorial plus X to the fourth over 4 factorial,

etcetera. If we wish to write this out in summation

notation, it would be the sum K goes from 0 to infinity X to the 2 K over 2 K

quantity factorial. Notice that just like the Taylor series

for cosine of X, COSH of X consists of the even powers but with no alternating

sign. That is another relationship between the

trigonometric and the hyperbolic trigonometric functions.

Does the same hold for the hyperbolic SINH.

Well let's investigate and see. Following the same method as before we

will use the Taylor series for E to the X.

And then the Taylor series for E to the minus X but now instead of adding these

two terms together, we are going to subtract the ladder from the former.

This leads to a cancellation of all the even powered terms and distributing the

minus sign through and adding, we obtain all of the odd degree terms in the Taylor

Series for E to the X. Thus the sum K goes from 0 to infinity.

X to the 2K plus 1 over 2K plus 1 quantity factorial.

Just like sign. We can continue our exploration of these

functions by proceeding as if the Taylor series are like long polynomials.

Hence computing things like integrals or derivatives can be done, term by term.

Let's consider what the derivative of the hyperbolic sine of X would be.

Well, we can differentiate the terms of the Taylor Series.

Since the derivative of X to the 2K plus 1 equals 2K plus 1 times X to the 2K.

We can see by dividing by 2K plus 1 quantity factorial and summing as K goes

from 0 to infinity. That, the derivative is the sum.

K goes from 0 to infinity of X to the 2K over 2K, quantity factorial.

That is simply the hyperbolic cosine of X.

In similarity to what happens with trig functions the derivative of sinh is cosh.

Likewise, what is the derivative of cosh of X?

If we differentiate term by term we can see that the derivative of X to the 2K is

2K times X to the 2K minus 1. Now we have to perform a shift in the

index, here. In order to avoid problems with what

happens when K equals 0. Re-indexing properly gives us the sum, K

goes from 0 to infinity of X to the 2 K plus 1 over 2 K plus 1 quantity

factorial. That is the hyperbolic sine of X.

And so we see that the hyperbolic trade functions are very nice.

The derivative of sinh is cosh. The derivative of cosh is sinh.

And this becomes clear from the Taylor series.