The rate of change of the function f(x) = 8 sec(x) + 8 cos(x) is given by the expression 8 sec(x) tan(x) − 8 sin(x). Show that this expression can also be written as 8 sin(x) tan2(x). Use the Reciprocal and Pythagorean identities, and then simplify.

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Step-by-step explanation:

Solution:-

- The given form for rate of change is:

                              8 sec(x) tan(x) − 8 sin(x).

- The form we need to show:

                              8 sin(x) tan2(x)

- We will first use reciprocal identities:

- Now take LCM:

- Using pythagorean identity , sin^2(x) + cos^2(x) = 1:

- Again use pythagorean identity tan(x) = sin(x) / cos(x):

Step-by-step explanation:

We are going to call g(x) the expression 8 sec(x) tan(X) - 8sin(x), so:

Now, we have the following identities:

1.

2.

So, if we replace that identities on the initial equation, we have:

Now, we need to sum both terms in the equation as:

Then, factoring 8sin(x), we get:

Now, we also have the following identity:

Finally, replacing on g(x), we get:

560

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