Respuesta :
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:
[tex]8\frac{sin(x)}{cos^2(x)} - 8sin(x)[/tex]
- Now take LCM:
[tex]8\frac{sin(x)- sin(x)*cos^2(x)}{cos^2(x)}[/tex]
- Using pythagorean identity , sin^2(x) + cos^2(x) = 1:
[tex]8*sin(x)*\frac{1- cos^2(x)}{cos^2(x)} = 8*sin(x)*\frac{sin^2(x)}{cos^2(x)}[/tex]
- Again use pythagorean identity tan(x) = sin(x) / cos(x):
[tex]8*sin(x)*tan^2(x)[/tex]
Answer:
[tex]8sec(x)tan(x)-8sin(x)=8sin(x)tan^2(x)[/tex]
Step-by-step explanation:
We are going to call g(x) the expression 8 sec(x) tan(X) - 8sin(x), so:
[tex]g(x)=8sec(x)tan(x)-8sin(x)[/tex]
Now, we have the following identities:
1. [tex]sec(x)=\frac{1}{cos(x)}[/tex]
2. [tex]tan(x)=\frac{sin(x)}{cos(x)}[/tex]
So, if we replace that identities on the initial equation, we have:
[tex]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)[/tex]
Now, we need to sum both terms in the equation as:
[tex]g(x)=8\frac{sin(x)}{cos^{2}(x)}-8sin(x)\\g(x)=\frac{8sin(x)-8sin(x)cos^{2}(x)}{cos^{2}(x)}[/tex]
Then, factoring 8sin(x), we get:
[tex]g(x)=8sin(x)*\frac{1-cos^2(x)}{cos^2(x)}[/tex]
Now, we also have the following identity:
[tex]sin^{2}(x) + cos^2(x)=1\\or\\sin^{2}(x) = 1-cos^{2}(x)[/tex]
Finally, replacing on g(x), we get:
[tex]g(x)=8sin(x)*\frac{sin^2(x)}{cos^2(x)}\\g(x)=8sin(x)*tan^2(x)[/tex]
