Question

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.

Answer 1

Answer:

Shown - See explanation

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:

                              $8\frac{sin(x)}{cos^2(x)} - 8sin(x)$

- Now take LCM:

                            $8\frac{sin(x)- sin(x)*cos^2(x)}{cos^2(x)}$

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

                            $8*sin(x)*\frac{1- cos^2(x)}{cos^2(x)} = 8*sin(x)*\frac{sin^2(x)}{cos^2(x)}$

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

                            $8*sin(x)*tan^2(x)$

Answer 2

Answer:

$8sec(x)tan(x)-8sin(x)=8sin(x)tan^2(x)$

Step-by-step explanation:

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

$g(x)=8sec(x)tan(x)-8sin(x)$

Now, we have the following identities:

1. $sec(x)=\frac{1}{cos(x)}$

2. $tan(x)=\frac{sin(x)}{cos(x)}$

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

$g(x)=(8\frac{1}{cos(x)}*\frac{sin(x)}{cos(x)})-8sin(x)\\g(x)=8\frac{sin(x)}{cos^{2}(x)}-8sin(x)$

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

$g(x)=8\frac{sin(x)}{cos^{2}(x)}-8sin(x)\\g(x)=\frac{8sin(x)-8sin(x)cos^{2}(x)}{cos^{2}(x)}$

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

$g(x)=8sin(x)*\frac{1-cos^2(x)}{cos^2(x)}$

Now, we also have the following identity:

$sin^{2}(x) + cos^2(x)=1\\or\\sin^{2}(x) = 1-cos^{2}(x)$

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

$g(x)=8sin(x)*\frac{sin^2(x)}{cos^2(x)}\\g(x)=8sin(x)*tan^2(x)$