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.

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:

- 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:

Step-by-step explanation:

let amount invested at 12%=x

amount invested in 10 %=y

amount invested in 8 %=z

z=2x

x+y+z=290,000

x+y+2x=290,000

3x+y=290,000  

multiply by 5

15x+5y=1,450,000  ... (1)

12/100 x+10/100 y+8/100 z=27,600

12x+10 y+8z=2,760,000

6x+5y+4z=1,380,000

6x+5y+4(2x)=1,380,000

14x+5y=1,380,000   ... (2)

(1)-(2) gives

x=1,450,000-1,380,000=70,000

z=2(70,000)=140,000

70,000+y+140,000=290,000

y+210,000=290,000

y=290,000-210,000=80,000

invested in 12 %=$ 70,000

invested in 10 %=$ 80,000

invested in 8 %=$ 140,000

(b)

annual income from 12 %=12/100 ×70,000= $8,400

annual income from 10 % =10/100×80,000=$ 9,600

annual income from 8 %=8/100×140,000= $11,200