Jorge has a credit card debt of $7,600.97. Jorge’s credit card has an APR of 23.99% and asks that Jorge makes a minimum payment of 3.5% of his balance. For Jorge’s current balance, what is the minimum payment he must make to the credit card company each month?

a.
$266.03

b.
$228.03

c.
$2,280.29

d.
$2,660.34


If Jorge (from #6) pays the minimum payment each month how many months will it take Jorge to pay off his balance, assuming he doesn’t make any more charges to his credit card? Assume the card compounds monthly. You may want to use the TVM solver

a.
About 43 months

b.
About 23 months

c.
About 29 months

d.
About 30 months

The complementary event to drawing a blue marble includes any outcome other than drawing a blue marble. Therefore, drawing a red marble, a green marble, a red or green marble, or not drawing a blue marble, all are complementary events.

In probability, the complementary event of an event represents all outcomes not covered by the original event. In this case, the original event is 'drawing a blue marble'. Thus, the complementary event would include any outcome other than drawing a blue marble.

Based on the options given:

All these are complementary events to drawing a blue marble, as they all represent outcomes other than 'drawing a blue marble'.

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Answer:

c. the time it takes to run the 100-meter race

Step-by-step explanation:

An independent variable is the variable which is not being controlled and it does not depend on the other variables. This is the variable which is being studied/measured in the experiment.

Option a. The previous running experience is not being considered and is not a variable under study in this case. So this is not the answer.

Option b. The teenagers who are being studied constitute the sample. These are not the variables.

Option c. Time is the independent variable, as it is being measured during the experiment and the conclusion is being drawn based on it.

Option d. Amount of caffeine is being decided by Cosella and is therefore not the independent variable.

Therefore, the correct answer is option c

Answer:

C'(4,4)

Step-by-step explanation:

The dilation of quadrilateral ABCD over the origin by a scale factor of 2 has the rule

(x,y)→(2x,2y)

So,

Hence, the coordinates of the image point C' are (4,4) (see attached diagram for details)

Answer:

The coordinates of C' = (4,4)

Step-by-step explanation:

The coordinates of C can be found by looking at the graph,

Coordinates of C = (2,2)

ABCD is dilated by a factor of 2 to get A'B'C'D'.

So, the coordinates of C' will be found by multiplying the coordinates of C by 2.

C' = (2*2,2*2)

C' = (4,4)

So, The coordinates of C' = (4,4)

Answer: The contrapositive statement :"If you do not have a gerbil, then you are not a pet owner.”

It is false.

Counter example : If you have a dog, then you are a pet owner.

Step-by-step explanation:

We know that the contrapositive of a statement of the form " If a then b" is "If not a then not b.

Given: The conditional statement is "If you have a gerbil, then you are a pet owner.”

Then the contrapositive statement will be "If you do not have a gerbil, then you are not a pet owner.”

The contrapositive statement is false.

Counter example : If you have a dog, then you are a pet owner.

Means if you area pet owner then it can be any pet not just gerbil.

Answer:

The contrapositive is false.

Step-by-step explanation:

Given conditional statement is :

If you have a gerbil, then you are a pet owner (conditional statement is in the form of if p, then q)

The contrapositive is represented as (if not q, then not p)

So, the contrapositive sentence will be :

If you are not a pet owner, then you do not have a gebril.

This statement is false as if you are not a pet owner then you will not have any animal with you and not only gebril.

Answer:

  (1, π/3 +2kπ), (-1, 4π/3 +2kπ) . . . where k is any integer

Step-by-step explanation:

Adding any multiple of 2π to the angle results in the same point in polar coordinates.

Adding 180° (π radians) to the point effectively negates the magnitude. As above, adding any multiple of 2π to this representation is also the same point in polar coordinates.

There are an infinite number of ways the coordinates can be written.

Answer:

All the polar coordinates of point P are [tex](1,2n\pi+\frac{\pi}{3})[/tex] and [tex](-1,(2n+1)\pi+\frac{\pi}{3})[/tex], where n is an integer.

Step-by-step explanation:

The given point is

[tex]P=(1,\frac{\pi}{3})[/tex]            .... (1)

If a point is defined as

[tex]P=(r,\theta)[/tex]          .... (2)

then the polar coordinates of point P is defined as

[tex](r,\theta)=(r,2n\pi+\theta)[/tex]

[tex](r,\theta)=(-r,(2n+1)\pi+\theta)[/tex]

where, n is an integer and θ is in radian.

From (1) and (2) we get

[tex]r=1, \theta=\frac{\pi}{3}[/tex]

So, the polar coordinates of point P are

[tex](r,\theta)=(1,2n\pi+\frac{\pi}{3})[/tex]

[tex](r,\theta)=(-1,(2n+1)\pi+\frac{\pi}{3})[/tex]

Therefore all the polar coordinates of point P are [tex](1,2n\pi+\frac{\pi}{3})[/tex] and [tex](-1,(2n+1)\pi+\frac{\pi}{3})[/tex], where n is an integer.

Answer:

21°

Step-by-step explanation:

the angles of triangle abc has to = 180 so subtract the known angles to get 21

angles bca and daf have to be the same so its 21

Answer:

[tex]x=21[/tex]

Step-by-step explanation:

We have been given an image of two parallel lines cut by a transversal. We are asked to find the measure of x.

We know that corresponding angles of two parallel lines are equal. We can see that angle BCA and angle DAF are corresponding angles.

Let us find measure of angle BCA using angle sum property.

[tex]54^{\circ}+105^{\circ}+m\angle BCA=180^{\circ}[/tex]

[tex]159^{\circ}+m\angle BCA=180^{\circ}[/tex]

[tex]159^{\circ}-159^{\circ}+m\angle BCA=180^{\circ}-159^{\circ}[/tex]

[tex]m\angle BCA=21^{\circ}[/tex]

Therefore, the value of x is 21 degrees.

Answer:

Please provide the polynomial to answer this question

Step-by-step explanation:

An expression will be said a polynomial if it contains variables like x, p , q, t etc etc.

They can be of any degree. like [tex]x^2[/tex] , [tex]y^3[/tex] , [tex]t^4[/tex] etc

The degree of a polynomial is the highest  exponent of the variable we have in the polynomial

Example : Degree of polynomial [tex]x^3-1 = 3[/tex]

If it do not have any variable , it is not called a polynomial because it basically gives us a constant

The given function violates both conditions for a polynomial, the correct answer is B. This is not a polynomial function because the variable powers are not all non-negative integers, and the coefficients are not constants.

To determine if the given function is a polynomial, we need to check two things:

The variable powers should be non-negative integers (whole numbers).

The coefficients of each term should be constants.

Let's analyze the function f(x)=√x+1 · (x+2)

Variable powers: The variable powers in the function are 1/2 and 1. The power 1/2 (square root) is not a non-negative integer, and thus violates the first condition for a polynomial.

Coefficients: The coefficients in the function are (1/2) and 1, which are not constants, but rather they involve variables. This also violates the second condition for a polynomial.

Since the given function violates both conditions for a polynomial, the correct answer is:

B. This is not a polynomial function because the variable powers are not all non-negative integers, and the coefficients are not constants.

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The complete question is:

Determine if the function is a polynomial function. If the function is a polynomial function, state the degree and the leading coefficient. If the function is not a polynomial, state why.

f(x)=√x+1 · (x+2)

A.This is not a polynomial function because there is no leading coefficient.

B. This is not a polynomial function because the variable powers are not all non-negative integers.

C. This is not a polynomial function because the factors are not all linear.

D. This is not a polynomial function because it is not written in the form f(x) = axⁿ + bxⁿ⁻¹ + ..... + rx² + sx + t.

Answer:

7

Step-by-step explanation:

the slope intercept form is y=mx + b

where b is the y intercept,

and your question is y=5x + 7

so the y intercept form would equal to 7

Answer:

C. √13

Step-by-step explanation:

The distance between two points is given by

d =sqrt( (x2-x1)^2 + (y2-y1)^2)

  = sqrt( (3-0)^2 + (-3--1)^2)

  = sqrt( 3^2 + (-3+1)^2)

  = sqrt( 9+(-2)^2)

  = sqrt( 9+4)

  = sqrt(13)

To answer this, you basically use Pythagoras' Theroem, but instead of:

[tex]c = \sqrt{a^{2} + b^{2}}[/tex]

it will be :

[tex]distance = \sqrt{(y - y1)^{2} + (x - x1)^{2}  }[/tex]

So you are finding the squareroot of the (difference in y coordinates)² plus (difference in x coordinates) ²:

x is the x-coordinate of (0, -1)      (so x = 0)

y is the y-coordinte of (0, -1)        ( so y = -1)

x1 is the x coordinate of (3, -3)        ( so x1 = 3)

y1 is the y coordinate of (3, -3)        (so y1 = -3)

--------------------------------------------------

Now, lets find the distance between the two points, by substituting all of this values into the equation at the top:

[tex]distance = \sqrt{(y - y1)^{2} + (x - x1)^{2}  }[/tex]    (substitute in values)

[tex]distance = \sqrt{( 0 -3)^{2} + (-1 - -3)^{2} }[/tex]   (simplify: note -1 - - 3 = -1 + 3)

[tex]distance = \sqrt{( -3)^{2} + (-1 +3)^{2} }[/tex]     (simplify)

[tex]distance = \sqrt{( -3)^{2} + (2)^{2} }[/tex]           (now square the numbers)

[tex]distance = \sqrt{9 + 4 }[/tex]                   (simplify)

[tex]distance = \sqrt{13 }[/tex]

___________________________________________

C.  [tex]\sqrt{13}[/tex]

Answer:  a. reject the null hypothesis, church attendance and marital status are dependent

Step-by-step explanation:

Given : A Chi square test has been conducted to assess the relationship between marital status and church attendance. The obtained Chi square is 23.45 and the critical Chi square is 9.488.

Null hypothesis : There is no relationship between the variables.

Alternative hypothesis : There is a relationship between the variables.

Here we can see that the obtained chi-square (23.45) value is greater than the critical chi square value (9.488) , then we have to reject the null hypothesis.

So the correct answer is reject the null hypothesis, church attendance and marital status are dependent.

Given the obtained Chi-square 23.45 is greater than the critical Chi square 9.488, we reject the null hypothesis, implying there is a significant relation or dependence between marital status and church attendance.

In a Chi square test, if the obtained Chi square value is higher than the critical Chi square value, it means that the observed data significantly deviates from what is expected under the null hypothesis. Therefore, in this case, where the obtained Chi square is 23.45 and the critical number is 9.488, we would reject the null hypothesis. Considering that the null hypothesis is generally posed under the assumption of no relation or independence between the variables being tested, rejecting it thus implies that there is a significant relationship or dependence between marital status and church attendance. Therefore, the correct answer to the question is a. reject the null hypothesis, church attendance and marital status are dependent

.

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Answer:

So a translation and a rotation and that is it.

D.

Step-by-step explanation:

There is no reflection.

If we had a negative in front of the x there would have been a reflection.

We can definitely tell the graph was translated 3 units up because of the +3.

There are other ways to look at a translation of a line but it has been translated for sure.

It has also been rotated because of the factor of 1/9 in front of x.

Let me show you a graph.

Comparing to y=mx+b where m is slope and b is y-intercept, we see the slope of f(x)=x is 1 and the y-intercept of f(x)=x is 0 while

the slope of g(x)=(1/9)x+3 is (1/9) and y-intercept of g(x)=(1/9)x+3 is 3.

I drew them here on the graph.

I also drew y=x+3 which has slope 1 and y-intercept 3.

So I just wanted to show you have we translate y=x up to y=x+3, you still don't have the same line.  But notice if you rotate y=x+3 using the (0,3) as the center of rotation you could get the green line to lay on the orange line.

Is this supposed to be multiple choice? If so it's a terrible question as the answer is all of the above. I'll sketch out the transformations, avoiding details.

Transform y=x to y=x/9 + 3

a. using rotation and reflection

Rotate around the origin to get y=-x/9 and reflect in y=3/2 to get y=x/9 +3

b. using translation, reflection, rotation

Translate to y = x - 3, reflect in the x axis to get y = x + 3 and rotate around (0,3) to get the result.

c. using translation and reflection.

Translate to y = x + 3 and reflect along the line which bisects the angle at (0,3).

d. rotation and translation

Rotate to a slope of 1/9 and translate up 3 units.

Answer:

51.496 miles

Step-by-step explanation:

circumference of circle= 2pir

here we will tale radius as 8.2 because 16.4 is diameter of circle.

2*3.14*8.2=

6.28*8.2=51.496 miles

[tex]6x[/tex]

............................................

Answer:

16. the expression below.

17. about 470,933 MD's and about 18,743 DO's

The expression for the total number of doctors is [tex]\frac{-1374t^2+43192.22t+67805.35}{(65-2t)(85-t) }[/tex]

a. MD + DO

[tex]\frac{28390+693t}{85-t} +\frac{776-12t}{65-2t}[/tex]

This can be simplified as

[tex]\frac{(28390+693t)(65-2t)+(776-12t)(85-t) }{(85-t)(65-2t) }[/tex]

[tex]= \frac{-1374t^2+43192.22t+67805.35}{(65-2t)(85-t) }[/tex]

b. If t = 0 in 1980, in 2010, t = 30

The number of MD would be

[tex]\frac{28390+693*30}{85-30} \\= 379[/tex]

The number of DO would be

[tex]\frac{776-12*30}{65-2*30}\\\\= 83[/tex]

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Answer:

[tex]z=0.923[/tex]

Step-by-step explanation:

we know that

A relationship between two variables, x, and z, represent an inverse variation if it can be expressed in the form [tex]z*x=k[/tex] or [tex]z=k/x[/tex]

step 1

Find the value of k

For x=6, z=2

[tex]z*x=k[/tex]

substitute

[tex]2*6=k[/tex]

[tex]k=12[/tex]

therefore

The equation of the inverse variation is equal to

[tex]z*x=12[/tex]

step 2

What is the value of z when x=13

substitute the value of x in the equation and solve for z

[tex]z*(13)=12[/tex]

[tex]z=12/13[/tex]

[tex]z=0.923[/tex]

Answer:

  10-2x

Step-by-step explanation:

If x is the length of one side of the square, the total width of the cardboard has two squares cut from it. The amount of width remaining is the width of the box. Since the original width of the cardboard is 10, the width of the box is ...

  10 -2x

Answer:

$98.47

Step-by-step explanation:

1. 2(9.5) + 3 (7.95) + 49.95 = $92.80

6% • $92.80 = $5.57

$92.80 + $5.57 = $98.47

Answer:

[tex]\large\boxed{\Phi=\dfrac{\pi}{9}+\dfrac{2k\pi}{3}\ or\ \Phi=-\dfrac{\pi}{9}+\dfrac{2k\pi}{3}}[/tex]

Step-by-step explanation:

[tex]\cos3\Phi=\dfrac{1}{2}\qquad\text{substitute}\ 3\Phi=\theta\\\\\cos\theta=\dfrac{1}{2}\iff\theta=\dfrac{\pi}{3}+2k\pi\ or\ \theta=-\dfrac{\pi}{3}+2k\pi\qquad k\in\mathbb{Z}\\\\\text{We're going back to substitution:}\\\\3\Phi=\dfrac{\pi}{3}+2k\pi\ or\ 3\Phi=-\dfrac{\pi}{3}+2k\pi\qquad\text{divide both sides by 3}\\\\\Phi=\dfrac{\pi}{9}+\dfrac{2k\pi}{3}\ or\ \Phi=-\dfrac{\pi}{9}+\dfrac{2k\pi}{3}[/tex]

[tex]\huge{\boxed{\frac{5}{2}}}[/tex]

Slope-intercept form is [tex]f(x)=mx+b[/tex], where [tex]m[/tex] is the slope and [tex]b[/tex] is the y-intercept.

[tex]\frac{5x}{2}=\frac{5}{2}x[/tex], so change the equation to represent this. [tex]f(x)=\frac{5}{2}x+3[/tex]

Great! Now the function is in slope-intercept form, so we can just see that [tex]m[/tex], or the slope, is equal to [tex]\boxed{\frac{5}{2}}[/tex]

Answer:

6 units

Step-by-step explanation:

A rectangle's width is one-fourth of its length.

w = 1/4 l

area is 9 square units

A = l*w = 9

  Replacing w with 1/4l

  l * (1/4l) = 9

1/4 l^2 = 9

Multiply each side by 4 to clear the fraction

4 * 1/4 l^2 = 4*9

l^2 = 36

Take the square root of each side

sqrt (l^2) = sqrt(36)

l =6

Answer:

6 units

Step-by-step explanation:

Let the length = L

Then the width = L/4

area = length * width

area = L * L/4 = 9

L^2/4 = 9

L^2 = 36

L = 6 or L = -6

Since we are dealing with a length, we eliminate the negative answer.

Answer: 6 units

P.S. Your equation is incorrect. L(L) = 9 would work for a square with 4 congruent sides and area 9. Here the sides have different lengths. The equation is L(L/4) = 9.

Answer:

[tex]m =-\frac{1}{2}[/tex]

Step-by-step explanation:

The equation of a line in the pending intersection form is:

[tex]y = mx + b[/tex]

Where m is the slope of the line and b is the intersection with the y axis.

In this case we have the following equation

[tex]y-3=-\frac{1}{2}(x-2)[/tex]

To find the slope of this line you must rewrite it in the form

[tex]y = mx + b[/tex]

Then we solve the equation for y.

[tex]y-3=-\frac{1}{2}(x-2)[/tex]

[tex]y=-\frac{1}{2}(x-2)+3[/tex]

[tex]y=-\frac{1}{2}x-2*(-\frac{1}{2})+3[/tex]

[tex]y=-\frac{1}{2}x+1+3[/tex]

[tex]y=-\frac{1}{2}x+4[/tex]

Note that [tex]m =-\frac{1}{2}[/tex]

Finally the slope is: [tex]m =-\frac{1}{2}[/tex]

The slope of the line with equation; y-3 = -1/2(x-2) is; slope, m = -1/2.

According to the question, the equation of the line in discuss is; y-3 = -1/2(x-2).

To determine the slope of the line, we need to rearrange the equation such that it resembles the slope-intercept form of the equation of a straight line as follows;

The equation of a straight line; y = mx + c.

Now, we expand the equation of the line and rearrange as follows;

By comparison, the slope of the line given bey the equation, y-3=-1/2(x-2) is; slope, m = -1/2.

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Answer:

[tex]\frac{7}{10}[/tex]

Step-by-step explanation:

Fraction of the amount donated to school library = [tex]\frac{2}{10}[/tex]

Fraction of the amount donated to animal shelter = [tex]\frac{1}{10}[/tex]

Fraction of the amount donated to food bank = [tex]\frac{4}{10}[/tex]

The rest of the amount was saved for next project.

Thus, the total fraction of the amount donated will be the sum of fractions of amount donated to school library, animal shelter and food bank.

i.e.

Fraction of the amount donated = [tex]\frac{2}{10}+\frac{1}{10}+\frac{4}{10} = \frac{7}{10}[/tex]

This means, Tyler and Katie donated [tex]\frac{7}{10}[/tex] of their profits.

Answer:−1.825

Step-by-step explanation:

Answer: 0.4758

Step-by-step explanation:

Given : Mean : [tex]\mu=137\text{ lb}[/tex]

Standard deviation : [tex]\sigma =28.9\text{ lb}[/tex]

Also, the new population of pilots has normally distributed .

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x=130 lb .

[tex]z=\dfrac{130-137}{28.9}=-0.2422145\approx-0.24[/tex]

For x=171lb.

[tex]z=\dfrac{171-137}{28.9}=1.1764705\approx1.18[/tex]

The p-value =[tex]P(-0.24<z<1.18)=P(z<1.18)-P(z<-0.24)[/tex]

[tex]=0.8809999-0.4051651=0.4758348\approx0.4758348\approx0.4758[/tex]

Hence, the required probability : 0.4758

Based on the properties of a parallelogram, the missing angle measures are:

The opposite angles of a parallelogram are defined as congruent angles, while the adjacent angles in a parallelogram are supplementary.

Thus:

m∠U = m∠S = 70°

m∠R = 180 - 70 = 110°

m∠T = m∠R =110°

Therefore, based on the properties of a parallelogram, the missing angle measures are:

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The missing angle measures are:

A parallelogram's opposing angles are known as congruent angles, whilst its adjacent angles are known as supplementary angles.

So, by the property of parallelogram

m∠U = m∠S = 70°

m∠R = 180 - 70

m∠R = 110°

and, m∠T = m∠R =110°

Thus, by the properties of a parallelogram, the angle are:

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Answer:

Option D. Divide 15 by 2

Step-by-step explanation:

we know that

To find the y-coordinate of the midpoint of a vertical line, adds the y-coordinates of the endpoints and divide by two

we have the endpoints

(0,0) and (0,15)

The y-coordinate of the midpoint is (0+15)/2=15/2=7.5

therefore

Divide 15 by 2

Answer:

the answer is to count by hands ( A ) and divide 15 by 2 ( D )

Step-by-step explanation:

Answer: 0.25

Step-by-step explanation:

Given : The  future lifetime of Gary is uniformly distributed with interval [0 years , 60 years].

Then the probability density function for Gary's future lifetime will be:-

[tex]f(x)=\dfrac{1}{60-0}=\dfrac{1}{60}[/tex]

The future lifetime of Eric is uniformly distributed with interval [0 years , 40 years].

Then the probability density function for Erin's future lifetime will be:-

[tex]f(x)=\dfrac{1}{40-0}=\dfrac{1}{40}[/tex]

Now, the joint density function for Gary and Eric's future lifetime :-

[tex]f(x,y)=f(x)f(y)=\dfrac{1}{40\times60}=\dfrac{1}{2400}[/tex]  [∵Their future lifetimes are independent. ]

Now, the probability that Gary dies first is given by :-

[tex]\int^{60}_{0}\int^{40}_{x}f(x,y)\ dy\ dx\\\\=\int^{60}_{0}\int^{40}_{x}\dfrac{1}{2400}\ dy\ dx\\\\=\int^{60}_{0}\dfrac{40-x}{2400}\ dx\\\\=\dfrac{1}{2400}[40x-\dfrac{x^2}{2}]^{60}_{0}\\\\=\dfrac{1}{2400}(2400-\dfrac{3600}{2})=0.25[/tex]

Hence, the probability that Gary dies first =0.25

Answer:

4 coconuts

Step-by-step explanation:

52-x/4+3+5

52-x/12

13-x/3

4

Answer:

5 coconuts

Step-by-step explanation:

If there were no loss , and 52 coconuts were distributed among 4 people, the number of coconuts each one would have obtained would be 13. Now as few coconuts are spoiled, the each one of those 4 people can get

1) 12 coconuts , and as per condition given there will be 3 more coconuts left. Hence total number of coconuts would be 12*4+3=48+3=51

Hence 1 coconuts would have been spoiled. Now let us see if this satisfies the second condition too.

If 51 coconuts are distributed equally among 3 people , each one gets 17 coconuts and none is left. But the condition says that 2 coconuts are left. Hence

Our assumption of loss of coconut was wrong.

2) 11 coconuts, as per condition given there will be 3 more coconuts left. Hence total number of coconuts would be 11*4+3=44+3=47

Hence 52-47=5 coconuts would have been spoiled. Now let us see if this satisfies the second condition too.

If 47 coconuts are distributed equally among 3 people , each one gets 15 coconuts and 2 coconuts are left. Hence it satisfies the second condition also. Let us see, if it satisfies third condition too.

If 47 coconuts are distributed among 5 people , each one gets, 9 cocnuts and coconuts left will be 2. Hence it satisfies the third condition also.

Hence , there were 5 coconuts spoiled.

Answer:

  63

Step-by-step explanation:

The two tangents to a circle from the same point are the same length.

  SD = SG

  5x +18 = 8x -9

  27 = 3x . . . . . . . . add 9-5x

  9 = x . . . . . . . . . . divide by 3

SG = 8x -9 = 8·9 -9

SG = 63

We have the following expression:

[tex]tan(\theta)cot(\theta)-sin^{2}(\theta)=cos^2(\theta)[/tex]

We know that:

[tex]cot(\theta)=\frac{1}{cot(\theta)}[/tex]

Therefore, by substituting in the original expression:

[tex]tan(\theta)\left(\frac{1}{tan(\theta)}\right)-sin^{2}(\theta)=cos^2(\theta) \\ \\ \\ Simplifying: \\ \\ 1-sin^2(\theta)=cos^2(\theta)[/tex]

We know that the basic relationship between the sine and the cosine determined by the Pythagorean identity, so:

[tex]sin^2(\theta)+cos^2(\theta)=1[/tex]

By subtracting [tex]sin^2(\theta)[/tex] from both sides, we get:

[tex]\boxed{cos^2(\theta)=1-sin^2(\theta)} \ Proved![/tex]

We have the following expression:

[tex]\frac{cos(\alpha)}{cos(\alpha)-sin(\alpha)}=\frac{1}{1-tan(\alpha)}[/tex]

Here, let's multiply each side by [tex]cos(\alpha)-sin(\alpha)[/tex]:

[tex](cos(\alpha)-sin(\alpha))\left(\frac{cos(\alpha)}{cos(\alpha)-sin(\alpha)}\right)=(cos(\alpha)-sin(\alpha))\left(\frac{1}{1-tan(\alpha)}\right) \\ \\ Then: \\ \\ cos(\alpha)=\frac{cos(\alpha)-sin(\alpha)}{1-tan(\alpha)}[/tex]

We also know that:

[tex]tan(\alpha)=\frac{sin(\alpha)}{cos(\alpha)}[/tex]

Then:

[tex]cos(\alpha)=\frac{cos(\alpha)-sin(\alpha)}{1-\frac{sin(\alpha)}{cos(\alpha)}} \\ \\ \\ Simplifying: \\ \\ cos(\alpha)=\frac{cos(\alpha)-sin(\alpha)}{\frac{cos(\alpha)-sin(\alpha)}{cos(\alpha)}} \\ \\ Or: \\ \\ cos(\alpha)=\frac{\frac{cos(\alpha)-sin(\alpha)}{1}}{\frac{cos(\alpha)-sin(\alpha)}{cos(\alpha)}} \\ \\ Then: \\ \\ cos(\alpha)=cos(\alpha).\frac{cos(\alpha)-sin(\alpha)}{cos(\alpha)-sin(\alpha)} \\ \\ \boxed{cos(\alpha)=cos(\alpha)} \ Proved![/tex]

We have the following expression:

[tex]\frac{cos(x+y)}{cosxsiny}=coty-tanx[/tex]

From Angle Sum Property, we know that:

[tex]cos(x+y)=cos(x)cos(y)-sin(x)sin(y)[/tex]

Substituting this in our original expression, we have:

[tex]\frac{cos(x)cos(y)-sin(x)sin(y)}{cosxsiny}=coty-tanx[/tex]

But we can also write this as follows:

[tex]\\ \frac{cosxcosy}{cosxsiny}-\frac{sinxsiny}{cosxsiny}=coty-tanx \\ \\ Simplifying: \\ \\ \frac{cosy}{siny}-\frac{sinx}{cosx} =coty-tanx \\ \\ But: \\ \\ \frac{cosy}{siny}=coty \\ \\ \frac{sinx}{cosx}=tanx \\ \\ Hence: \\ \\ \boxed{coty-tanx=coty-tanx} \ Proved![/tex]

We have the following expression:

[tex]\ln\left|1+cos \theta\right|+\ln\left|1-cos \theta\right|=2\ln\left|sin \theta\right|[/tex]

By Logarithm product rule, we know:

[tex]log_{b}(x.y) = log_{b}(x) + log_{b}(y)[/tex]

So:

[tex]\ln\left|1+cos \theta\right|+\ln\left|1-cos \theta\right|=\ln\left|(1+cos \theta)(1-cos \theta)\right|[/tex]

The Difference of Squares states that:

[tex]a^2-b^2=(a+b)(a-b) \\ \\ So: \\ \\ (1+cos \theta)(1-cos \theta)=1-cos^2 \theta[/tex]

Then:

[tex]\ln\left|(1+cos \theta)(1-cos \theta)\right|=\ln\left|1-cos^{2} \theta\right|[/tex]

By the Pythagorean identity:

[tex]sin^2(\theta)+cos^2(\theta)=1 \\ \\ So: \\ \\ sin^2 \theta = 1-cos^2 \theta[/tex]

Then:

[tex]\ln\left|1-cos^{2} \theta\right|=\ln\left|sin^2 \theta|[/tex]

By Logarithm power rule, we know:

[tex]log_{b}(x.y) = ylog_{b}(x)[/tex]

Then:

[tex]\ln\left|sin^2 \theta|=2\ln\left|sin \theta|[/tex]

In conclusion:

[tex]\boxed{\ln\left|1+cos \theta\right|+\ln\left|1-cos \theta\right|=2\ln\left|sin \theta\right|} \ Proved![/tex]