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trapecia [35]
3 years ago
6

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 t

his expression can also be written as 8 sin(x) tan2(x). Use the Reciprocal and Pythagorean identities, and then simplify.
Mathematics
2 answers:
liq [111]3 years ago
8 0

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)

Ghella [55]3 years ago
3 0

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)

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