Ruth Barr rented a car for 5 days at 59.95 per day with unlimited mileage she drove 1156 miles and paid 137.76 for gasoline. What was the total cost per mile to rent the car

Answer:

A.  5 seconds

Step-by-step explanation:

This quadratic (second degree polynomial) models parabolic motion.  The vertex of the function is the projectile's max height.  When the height of the object is 0 is when it hits the water.  Set the function equal to 0 then and factor.  I used the quadratic formula on my calculator, but if you need to, stick it into the quadratic formula and do it long hand.  The values of x (which is actually time here) are -1 and 5.  The two things in math that will never ever be negative are distance measurements and time, so we can disregard the -1 and say that it takes 5 seconds for the rocket to hit the water.

It would take 5 seconds for the rocket to hit the lake.

An equation is used to show the relationship between two or more variables and numbers.

Let g(x) represent the height of the rocket at time x seconds. Given that:

g(x)= -16x² + 64x + 80

The rocket touches the lake, when the height is 0. Hence:

0 = -16x² + 64x + 80

x = 5 seconds.

It would take 5 seconds for the rocket to hit the lake.

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Answer:

x = 74°

Step-by-step explanation:

The angle whose vertex lies on the circle, that is angle x is one- half the measure of it's intercepted arc, that is

x = 0.5 × 148° = 74°

Answer:

D. 46

Step-by-step explanation:

Put 12 where n is, then do the arithmetic.

t12 = 4·12 -2 = 48-2 = 46

Answer:

48.3 cm²

Step-by-step explanation:

The area (A) of the yellow region = area of square - area of quarter circle, that is

A = 15² - ([tex]\frac{1}{4}[/tex] πr² )

  = 225 -( [tex]\frac{1}{4}[/tex] × π × 15² )

  = 225 - 176.71 ≈ 48.3

Answer:8 ounces of oysters per minute

Step-by-step explanation:

Answer:

5 ounces of oysters per minute

Step-by-step explanation:

We are given a graph of the total weight, in ounces, of Steve's bag of oysters, y, with respect to the amount of time that he spends looking for them in minutes, x,

Now we are supposed to find the average rate of change over the interval [2, 10]

Formula of Average rate : [tex]\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]

Now At x = 2 , f(2)= y = 20

At x=10  , f(10)= y = 60

Substitute the values in the formula

Average rate : [tex]\frac{f(10)-f(2)}{10-2}[/tex]

Average rate : [tex]\frac{60-20}{10-2}[/tex]

Average rate : [tex]\frac{40}{8}[/tex]

Average rate : [tex]5[/tex]

Thus he average rate of change over the interval [2, 10]  is 5 ounces of oysters per minute

Answer:

The percentage of the people who live in the forests that are indigenous is [tex]20\%[/tex]

Step-by-step explanation:

we know that

To find the percentage of the people who live in the forests that are indigenous, divide the number of people that are indigenous by the total number of people that live in forests

so

[tex]\frac{60}{300} =0.2[/tex]

Convert to percentage

Multiply by 100

[tex]0.2*100=20\%[/tex]

Answer: INTRODUCTION

"Of all the environmental impacts of the study projections, deforestation probably poses the most serious problems for the world, particularly for the developing world."

Global 2000

"It has been predicted that within the next 25-30 years, most of the humid tropical forest as we know it, will be transformed into unproductive land, and the deterioration of the savannah into desert will continue at ever-increasing speed."

As you read this sentence, 50 to 100 acres of primary tropical forests will be eliminated, disrupted, degraded or impoverished. Yearly, an area of tropical forest the size of Great Britain is "converted" from an area equal to the size of Europe.

If present trends continue, by the year 2000, all tropical forests, with the exception of two areas - the western Brazilian Amazon and Central Africa - will have been destroyed. Since 1950, according to the U.N. Food and Agriculture Organization (FAO), half of the world's forests have disappeared. Latin America has lost 37 percent of its tropical forests; Central America, 66 percent; Southeast Asia, 38 percent; Central Africa, 52 percent. Nearly 20 million acres are destroyed annually.

As areas of tropical forests are destroyed or degraded, tribal groups are forced to change their resource base. In some cases they move into areas occupied by other groups, straining the area's resources. In other cases they are forced to relocate outside of forests, permanently altering their way of life by converting to agriculture or to cash employment. Rarely are the rights of these groups to the lands they occupy recognized. Further, their intimate knowledge of the area's resources and how to manage them are nearly always ignored.

Millions of indigenous people live in tropical moist forests which cover some 3.6 million square miles in 70 countries. More than 80 percent of these forests are found in Bolivia, Brazil, Colombia, Gabon, Indonesia, Malaysia, Peru, Venezuela, and Zaire, while 30 additional countries contain sufficient tracts to have significant ecological and biotic values. If these areas are to be managed effectively into the next century, the indigenous peoples that inhabit them should be consulted.

Use trigonometry to find the ramp distance.

sin(4.76°) = 2/r

Let r = ramp distance in feet

r = 2/sin(4.76°)

r = 24.1015718106

Round off to the nearest foot.

Doing so we get r = 24 feet.

The ramp is 24.0 feet long.

Answer: Choice C

The answer is:

The man did consume two standard drinks in two hours.

According to the standard drink conversion, we know that each 12 ounce beer we have one (1) standard drink, and each 1.5 ounce shot of liquor, we have another standar drink.

Calculating we have:

[tex]DrinksConsumed=12Oz(Beer)+1.5Oz(Liquor)=1StandardDrink+1StandardDrink\\\\DrinksConsumed=1StandardDrink+1StandardDrink=2StandardDrinks[/tex]

We have that the man did consume two standard drinks in two hours.

Have a nice day!

A man who consumed two 12 ounce beers and a 1.5 ounce shot of liquor has consumed three standard drinks according to the National Institute on Alcohol Abuse and Alcoholism's definition of a standard drink in the United States.

According to the National Institute on Alcohol Abuse and Alcoholism (NIAAA), a standard alcoholic drink in the United States is equivalent to 14 grams (0.6 ounces) of pure alcohol. This typically is found in a 12-ounce beer, a 5-ounce glass of wine, or a 1.5 ounce shot of distilled spirits or liquor. The man in the question consumed two 12 ounce beers and one 1.5 ounce shot of liquor. Given that both the beer and the shot each lay within the standard drink size, we can conclude that the man consumed three standard drinks in total. Alcohol consumption and understanding standard drinks is important to monitor one's drinking habits for overall health.

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Answer:

5 meters

Step-by-step explanation:

If the perimeter of the rectangle on the right is 24 m, and the length is 8, the width has to be 4, since 8+4+8+4=24. Since the scale factor is 24÷30=0.8 or 4/5, and the width of the rectangle on the right is 4, 4÷0.8=5, which is the width of the rectangle on the left.

Hope this helps! (P.S. I got it right on Edgenutiy, spelled wrong on purpose).

The width of the original rectangle is 5m.

The perimeter of a shape can be described as the path or boundary that surrounds it .

Perimeter of Rectangle = 2 (L + B)

L = length of rectangle

B = breadth of rectangle

Perimeter of the reduced rectangle = 24m

Let the length of original rectangle be L1

Let the breadth of original rectangle be B1

Let L1 = 8m, B1 = 4m

P = 2( L1 + B1) = 2(8 + 4) = 24m

Scale factor = P(Original Rectangle) / P(reduced rectangle)

Scale factor = 30 / 24 = 1.25

So, width of original rectangle = Scale factor * reduced width

Width of original rectangle = 1.25 * 4 = 5m

Hence, the width of the original rectangle is 5m.

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Answer:

Imagination is more important than knowledge. --Clifford big red dog

Step-by-step explanation:

a) To find the average cost per pair of shoes, we need to divide the total cost by the number of pairs produced. Since the manufacturer produced x pairs of shoes, the total cost is 12,000+19x dollars. Therefore, the average cost per pair is given by:

Average cost per pair = (12,000+19x) / x

b) Using the given information that this month, the manufacturer produced 1000 more pairs of shoes than last month, we can represent the number of pairs produced last month as x−1000. Since the average cost per pair dropped by $0.43, the equation representing the situation is:

(12,000+19(x)−43(x)) / x=Average cost per pair

Solving this equation will provide us with the value of x.

c) Yes, there is a mathematical restriction on the domain. Since the number of pairs produced cannot be negative (it doesn't make sense to produce a negative number of shoes), the domain must be restricted to x>0.

d) In the context of the problem, the reasonable domain would be x>0 since the number of pairs of shoes produced cannot be negative. Additionally, since the manufacturer produced 1000 more pairs of shoes this month than last month, it's reasonable to assume that the number of pairs produced last month is also positive, leading to a positive change of 1000 pairs. Therefore, the domain would be x>1000.

a) The expression for the average cost per pair of shoes is given by:

Average cost per pair = (12,000+19x) / x

To represent the situation, we use this expression to equate the average cost per pair before and after the change in production.

b) Substituting x+1000 for x in the expression, we get:

(12,000+19(x+1000)) / (x+1000) = Average cost per pair

We solve this equation to find the value of x.

c) Yes, there's a restriction on the domain. Since the number of pairs of shoes cannot be negative, the domain must be x>0.

d) Considering the context, the reasonable domain is x>1000, as the manufacturer produced 1000 more pairs this month than last month. Also, x must be positive, so x>0. Thus, the domain is x>1000.

Complete Question:

The cost in dollars to manufacture x pairs of shoes is given by 12,000 + 19x. This month, the manufacturer produced 1000 more pairs of shoes than last month. The average cost per pair dropped by $0.43.

a) Write an expression for the average cost per pair of shoes. Use this expression to write an equation to represent their situation.

b) Solve your equation

c) Are there any mathematical restrictions on the domain? Explain.

d) Determine reasonable domain in the context of the problem. Use your answers to parts I and II to answer the question.

Answer:

Step-by-step explanation:

first fit:

115 -> 300

500-> 600

358 -> 750

200 -> 350

375 -> not able to allocate  

Best fit:

115 -> 125

500 -> 600

358 -> 750

200 -> 200

375 -> not able to allocate

worst fit:

115 -> 750

500 -> 600

358 -> not able to allocate

200 -> 350

375 -> not able to allocate

Answer:

1 m 79 cm

Step-by-step explanation:

Subtract 1.46 m from 3.25 cm:  1.79 cm, or 1 m 79 cm

Answer: 7. y = 2 sin (2x) - 3

              [tex]\bold{8.\quad y=sin\bigg(8x + \dfrac{8}{\pi}\bigg)}[/tex]

Step-by-step explanation:

The general form of a sine equation is: y = A sin (Bx - C) + D    where

7. Given:

--> y = 2 sin (2x) - 3

8. Given:

[tex]\implies y=sin\bigg(8x + \dfrac{8}{\pi}\bigg)[/tex]

Answer:

(t, t -1, t)

Step-by-step explanation:

You have three unknowns but only 2 equations, so you can't really SOLVE this...you can get a solution with a variable still in it (I forget what this is called.  I think it refers to infinite many solutions).  Here's how it works:

Set up your matrix:

[tex]\left[\begin{array}{ccc}1&-2&1\\2&-1&-1\\\end{array}\right] \left[\begin{array}{ccc}2\\1\\\end{array}\right][/tex]

You want to change the number in position 21 (the 2 in the scond row) to a 0 so you have y and z left.  Do this by multiplying the top row by -2 then adding it to the second row to get that 2 to become a 0.  Multiplying in a -2 to the top row gives you:

[tex]\left[\begin{array}{ccc}-2&4&-2\\2&-1&-1\\\end{array}\right]\left[\begin{array}{ccc}-4\\1\\\end{array}\right][/tex]

Then add, keeping the first row the same and changing the second to reflect the addition:

[tex]\left[\begin{array}{ccc}-2&4&-2\\0&3&-3\\\end{array}\right] \left[\begin{array}{ccc}-4\\-3\\\end{array}\right][/tex]

The second equation is this now:

3y - 3z = -3.  Solving for y gives you y = z - 1.  Let's let z = t (some random real number that will make the system true.  Any number will work.  I'll show you at the end.  Just bear with me...)

lf z = t, and if y = z - 1, then y = t - 1.  So far we have that y = t - 1 and z = t.  Now we solve for x:

From the first equation in the original system,

x - 2y + z = 2.  Subbing in t - 1 for y and t for z:

x - 2(t - 1) + t = 2.  Simplify to get

x - 2t + 2 + t = 2  and  x - t = 0, and x = t.  So the solution set is (t, t - 1, t).  Picking a random value for t of, let's say 2, sub that in and make sure it works.  If:

x - 2y + z = 2, then t - 2(t - 1) + t = 2 becomes t - 2t + 2 + t = 2, and with t = 2, 2 - 2(2) + 2 + 2 = 2.    Check it:  2 - 4 + 4 = 2 and 2 = 2.  You could pick any value for t and it will work.

assuming 10 inches is the diameter

Answer:

≈

79

square inches.

Explanation:

The area of a circle is given by formula:

π

r

2

Where,

π

has a constant value of

3.14

and

r

denotes the radius.

The radius of the circle is , half the diameter =

d

2

=

10

2

=

5

inches

The area

=

π

r

2

=

3.14

×

(

5

)

2

=

3.14

×

(

25

)

=

78.5

square inches.

≈

79

square inches

The area of the ten-inch service plate is approximately 78.54 square inches.

To calculate the area of a circle, we use the formula:

Area = π * r²

where:

π (pi) is a mathematical constant approximately equal to 3.14159

r is the radius of the circle

Given:

Diameter = 10 inches

Finding the radius:

Radius = Diameter / 2 = 10 inches / 2 = 5 inches

Calculating the area:

Area = π * (5 inches)² ≈ 3.14159 * 25 square inches ≈ 78.54 square inches

The domain are the x values.

The problem is saying the x value is all numbers between 4 and 8

Replace x in the equation with 4 and 8 and solve to find the range for the y values:

y =4(4) -1 = 16-1 = 15

y = 4(8) - 1 = 32-1 = 31

So Y would be between 15 and 31.

The first answer is the correct one.

Answer:

Step-by-step explanation:

The product of two even numbers will have at least two factors of 2, so will be even: even numbers are closed under multiplication.

The product of two odd numbers will be an even number plus 1, so will be odd: odd numbers are closed under multiplication.

The product of two cubes will be the cube of the product of their cube roots: cubes are closed under multiplication.

__

The product of two negative numbers will be positive, so the set of negative numbers is not closed under multiplication.

Answer:

40.82 is the surface area of this solid.

For this case we have that the surface area of the figure is given by the surface area of a cone plus the surface area of a cylinder.

The surface area of a cylinder is given by:

[tex]SA = 2 \pi * r * h + 2 \pi * r ^ 2[/tex]

Where:

h: It's the height

A: It's the radio.

Substituting the values:

[tex]SA = 2 \pi * 1 * 3 + 2 \pi * 1 ^ 2\\SA = 6 \pi + 2 \pi * 1\\SA = 8 \pi\\SA = 25.12 \ units ^ 2[/tex]

On the other hand, the surface area of a cone is given by:

[tex]SA = \pi * r * s + \pi * r ^ 2[/tex]

Where:

A: It's the radio

s: inclination

Substituting the values:

S[tex]A = \pi * 1 * 6 + \pi * 1 ^ 2\\SA = 6 \pi + \pi\\SA = 7 \pi[/tex]

[tex]SA = 21.98 \ units ^ 2[/tex]

Thus, the surface area of the figure is:

[tex]47.1 \ units ^ 2[/tex]

ANswer:

[tex]47.1 \ units ^ 2[/tex]

Answer:

$148.21

Step-by-step explanation:

A suitable financial calculator, web site, or spreadsheet can figure this for you. Or you can use the formula given in your reference material (text or web site).

Answer:

Michelle's monthly payment will be $148.21.

Step-by-step explanation:

The EMI formula is =

[tex]\frac{p\times r\times(1+r)^{n} }{(1+r)^{n}-1}[/tex]

Here,

p = $10125

r = [tex]12.5/12/100=0.010417[/tex]

n = [tex]10\times12=120[/tex]

Putting all these values in the formula we get,

[tex]\frac{10125\times0.010417\times(1+0.010417)^{120} }{(1+0.010417)^{120}-1}[/tex]

=>[tex]\frac{10125\times0.010417\times(1.010417)^{120} }{(1.010417)^{120}-1}[/tex]

=$148.21

So, Michelle's monthly payment will be $148.21.

Answer:

In the figures attached, the complete question is shown.

What is the difference between a minor arc and a major arc?

the measure of a minor arc is less than 180°

the measure of a major arc is greater than 180°

How many letters do we use to name a MINOR arc? 2

How many letters do we use to name a MAJOR arc? 3

How many degrees are in a semi-circle? 180°

How many letters to name a SEMI CIRCLE? 3

1. Name of arc: AB Type of arc: minor

2. Name of arc: ADB Type of arc: major

3. Name of arc: PSQ  Type of arc: semi-circle

4. AE: minor

5. AEB: semi-circle

6. FDE: semi-circle

7. DFB: major

8. FA: minor

9. BE: minor

10. BDA: semi-circle

11. FBD: major

12. PQ and ST

13. QPT and  PUS

14. PUT and QPU

15. There are 360° degrees in a circle.

16. There are 180° degrees in a semi-circle.

17. The measure of the arc is equal to the measure of the central angle.

18. mPQ: 50°, mPXQ: 310°

19. mPQ: 90° , mPRQ: 270°

20. mPQ: 150° , mPXQ: 210°

21. mQS: 45°, mQRS: 315°

22. mGH: 30°, mGFH: 330°

23. mAB: 75°, mADB: 285°

Answer:

There are 13 boys in Mr. Marshals class

Step-by-step explanation:

Hello there! Mr. Marshal has 9 boys.

To find the number of boys Mr. Marshal has, start by finding the ratio. If there are 6 boys and 14 girls, the boys to girls ratio is 6:14. Simplified, the ratio is 3:7. So, if there are 21 girls, you want to find how many boys there are. To find this, find what you need to multiply 7 by to get 21 by and multiply 3 by that number.

21/7 = 3.

So, now we multiply 3 by 3 to get the number of boys.

3 x 3 = 9.

This means there are 9 boys, with a ratio of 9:21. If we simplify this, we get 3:7, making this answer correct.

I hope this helps and have a great day!

Answer:

  down

Step-by-step explanation:

Subtracting 1 from the y-coordinate moves a point down 1 unit. You know this because you know that the y-coordinate tells you the number of units the point is above the x-axis.

Every point on the graph of f(x) has coordinates (x, f(x)). If you subtract 1 from the y-coordinate, you have (x, f(x) -1) = (x, g(x)). The graph of this is a graph of f(x) that is 1 unit down from its original position.

In the equation g(x)=f(x)-1, g(x) translates the original function, f(x), one unit downward. This is an instance of vertical translation in mathematics.

In the equation g(x)=f(x)-1, the function g(x) is a translation of the function f(x). This process is specifically called a vertical translation. The -1 in the function g(x) = f(x) - 1 implies that the translation is downward. Therefore, g(x) translates the function f(x) 1 unit downward.

To further illustrate, let's consider the function f(x) = x^2 and its translation g(x) = x^2 - 1. If we graph both of these functions, you can observe that the graph of g(x) is exactly the same as the graph of f(x), but it is shifted one unit downward. This is the meaning of vertical translation in mathematics.

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Answer:

One

Step-by-step explanation:

4x + 5 = 10

Subtract 5 from both sides.

4x = 5

Divide both sides by 4.

x = 5/4

There is one solution.

Jackie's total spending on 2 packages of paper and 4 notebooks is represented by the equation 2(5.80) + 4d = 32, where d stands for the cost per notebook.

The student is looking to create an equation to represent the cost of Jackie's purchases of paper packages and notebooks. The given information is that Jackie bought 2 packages of papers for $5.80 each and 4 notebooks for d dollars each, and in total, she spent $32. To formulate the equation, we can use the following expression:

We know the cost of the paper packages is 2 multiplied by $5.80, and the cost of the notebooks is 4 multiplied by d dollars.

So the equation using D that represents the situation is:

Where d represents the cost of each notebook, and solving for d would give us the price per notebook.

Answer:

b.  .897x - 3.790 (that's in slope-intercept form; same thing as putting the y-intercept first)

Step-by-step explanation:

On your calculator, hit the "stat" button.  Hit #4 to "ClrList", then hit 2nd and the number 1 to clear the list in L1.  Then do the same to clear the list in L2.  Hit #4, then hit 2nd and the number 2 to clear the list in L2.

Hit "stat" then 1 (edit) and you'll be at your table to enter the values.  Use the left arrow key if needed to make sure you're entering your first values under column L1, which is our x values.  Enter one at a time, hitting "enter" after each, even the last one.  Then arrow over to the right and enter the y values one at a time under L2.  Hit enter after each, even the last one.  When you're done, hit "stat" again and arrow over to "calc".  If you have a TI 83, choose "4:LinReg" and hit "enter".  If you have a TI 84+, you will have to arrow down to choose "calculate".  When the word "calculate" is highlighted, hit enter and you'll have your equation!

  Equation of the regression line for the data given will be represented by the equation given in option B.  

    By using the utility for the regression line of the given data, equation of the regression line will be,

y = 0.89745x - 3.79005

y = 0.897x - 3.79

Or y = -3.79 + 0.897x

   Therefore, equation of the regression line given in option B will be the answer.

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Answer:

Part 1) The ratio of the perimeters of the larger figure to the smaller figure is equal to [tex]\frac{16}{13}[/tex]

Part 2) The ratio of the areas of the larger figure to the smaller figure is [tex]\frac{256}{169}[/tex]

Step-by-step explanation:

Part 1) What is the ratio of the perimeters of the larger figure to the smaller figure?

we know that

If two figures are similar, then the ratio of its corresponding sides is proportional, and this ratio is called the scale factor

Let

z-----> the scale factor

In this problem

The scale factor is equal to

[tex]z=\frac{32}{26}[/tex]

Simplify

[tex]z=\frac{16}{13}[/tex]

Remember

If two figures are similar, then the ratio of its perimeters is equal to the scale factor

so

The ratio of the perimeters of the larger figure to the smaller figure is equal to [tex]\frac{16}{13}[/tex]

Part 2) What is the ratio of the areas of the larger figure to the smaller figure?

we know that

If two figures are similar, then the ratio of its areas is equal to the scale factor squared

Let

z-----> the scale factor

we have

[tex]z=\frac{16}{13}[/tex]

so

[tex]z^{2}=(\frac{16}{13})^{2}=\frac{256}{169}[/tex]

therefore

The ratio of the areas of the larger figure to the smaller figure is [tex]\frac{256}{169}[/tex]

Answer:

  y = –4x + 5

Step-by-step explanation:

We can put the offered choices into standard form and compare.

y = –4x + 5 ⇒ 4x +y = 5 . . . . . intersecting line; one solution

y = 4x – 5 ⇒ 4x -y = 5 . . . . . .  same line, infinite solutions

2y = 8x – 10 ⇒ 4x -y = 5 . . . .  same line, infinite solutions

–2y = –8x – 10 ⇒ 4x -y = -5 . . . . parallel line, no solutions

Answer:

IS A: y = –4x + 5

Hope this helps!

Answer:

157 cm²

Step-by-step explanation:

volume = (1/3) * π * r² * h

volume = (1/3) * π * 5² * 6

volume = (1/3) * π * 25 * 6

volume = (1/3) * π * 150

volume = (1/3) * 471

volume = 157 cm²

Answer:

157 cubic centimeters

Step-by-step explanation:

Answer:

There is no clear relationship between the number of times students arrive late and the distances they live from school.

Step-by-step explanation:

By looking at the graph, there is no clear direction or association that the graph has. Therefore, there is no clear relationship between the variables being compared.

Answer:

looking at the graph, there is no clear direction or association

Step-by-step explanation:

Answer:

the answer is the letter a) -sin x

Step-by-step explanation:

Simplify the expression.

sine of x to the second power minus one divided by cosine of negative x

(1−sin2(x))/(sin(x)−csc(x))

sin2x+cos2x=11−sin2x=cos2x

cos2(x)/(sin(x)−csc(x))csc(x)=1/sin(x)cos2(x)/(sin(x)− 1/sin(x))= cos2(x)/((sin2(x)− 1)/sin(x))sin2(x)− 1=-cos2(x)cos2(x)/(( -cos2(x))/sin(x))

=-sin(x)

Answer:

[tex]-cos \ x[/tex]

Step-by-step explanation:

First of all, we must have to understand what is the described expression in the paragraph

"sine of x to the second power minus one divided by cosine of negative x"

In this sentence, we need to identify what are the elements and operations involved in the expression.

In the sentence appears ""to the second power", "minus" and "divided by" (highlighted)

"sine of x to the second power minus one divided by cosine of negative x"

Therefore, the expression must has three operations:

Now, we can identify what are the elements: "sine of x", "one" and "cosine of negative x"

Therefore, the expression is:

[tex]\frac{(sin\ x)^{2}-1}{cos(-x)}[/tex]

In order to find the simplified expression, we must have to apply these trigonometric identities:

Applying the first and third identities, we have:

[tex]\frac{(sin\ x)^{2}-1}{cos(-x)}=\frac{sin\x^{2}x-1}{cos\ x}[/tex]

From the second trigonometric identity, we have:

[tex]cos\x^{2}x=\ 1-sin\x^{2}x[/tex]

Now, multiplying by -1 in both sides:

[tex](-1)(cos\x^{2}x)=(-1)(1-\ sin\x^{2}x)[/tex]

In the left side, multiplying by -1 the sign of the expression changes:

[tex](-1)(cos\x^{2}x)=-cos\x^{2}x[/tex]

In the right side, multiplying by -1 changes the order of the substraction:

[tex](-1)(1-\ sin\x^{2}x)=\ sin\x^{2}x-1[/tex]

Putting all together:

[tex]-cos\x^{2}x=\ sin\x^{2}x-1[/tex]

Now, replacing values we have:

[tex]\frac{sin\x^{2}x-1}{cos\ x}=\frac{-cos\x^{2}x}{cos\ x}=-\frac{cos\x^{2}x}{cos\ x}[/tex]

Finally, the property of the first trigonometric identity (property of exponentiation) can be apply in this case:

[tex]-\frac{cos\x^{2}x}{cos\ x}=-\frac{(cos\ x)^{2}}{cos\ x}=-cos\ x[/tex]