The rate, in hundreds of people per hour, that the amount of people in an amusement park is changing is modeled by the function [tex]R(t)[/tex] attached where [tex]t=0[/tex] corresponds to 12 p.m.

If there are [tex]300[/tex] people in the park at 12 p.m., use a midpoint Riemann sum with three equal subintervals to approximate the total number of people, in hundreds, in the park at 6 p.m.

(a) To calculate the probability of the first miss occurring after 12 attempts, we need to consider the scenario in which the shooter makes the first 11 shots and then misses the 12th shot. The probability of making a free throw is given as p = 0.9.

The probability of making a shot is 0.9, so the probability of missing a shot is 1 - 0.9 = 0.1. Therefore, the probability of making 11 shots in a row is (0.9)^11.

The probability of missing the 12th shot is 0.1. Since these events are independent, we can multiply the probabilities together. Therefore, the probability of making the first 11 shots and missing the 12th shot is (0.9)^11 * 0.1.

Therefore, the probability of having the first miss after 12 attempts is (0.9)^11 * 0.1.

(b) To calculate the probability that the third miss occurs on the 30th attempt, we need to consider the scenario in which the shooter makes the first 29 shots and then misses the 30th shot.

The probability of making a shot is 0.9, so the probability of missing a shot is 1 - 0.9 = 0.1. Therefore, the probability of making 29 shots in a row is (0.9)^29.

The probability of missing the 30th shot is 0.1. Since these events are independent, we can multiply the probabilities together. Therefore, the probability of making the first 29 shots and missing the 30th shot is (0.9)^29 * 0.1.

However, we also need to consider that the shooter must miss the first two shots before reaching the 30th attempt. The probability of missing two shots in a row is (0.1)^2.

Therefore, the probability that the third miss occurs on the 30th attempt is (0.9)^29 * 0.1 * (0.1)^2.

Note that these calculations assume that each shot is independent of the others and that the shooter's probability of making a shot remains constant throughout the attempts.

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The 95% confidence interval for the population mean μ is (18.3, 21.7).

Given: n = 100, σ2 = 64, X-bar = 20, and the desired level of confidence is 95%.

We can use the formula for the confidence interval for the population mean when the population standard deviation is known:

CI = X-bar ± Zα/2 * σ/√n

where CI is the confidence interval, Zα/2 is the critical value from the standard normal distribution for the given level of confidence, σ is the population standard deviation, and n is the sample size.

Since the level of confidence is 95%, we have α = 0.05 and Zα/2 = 1.96.

Substituting the given values, we get:

CI = 20 ± 1.96 * 8/√100

Simplifying the expression, we get:

CI = (18.3, 21.7)

Therefore, we can be 95% confident that the true population mean μ lies between 18.3 and 21.7 based on the given sample data.

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Answer:

(X+8) (x+5)

Step-by-step explanation:

factorise it

The set of all points in a plane that are closer to point P than to point Q is the interior of a circle.

To understand why this is the case, imagine drawing two circles centered at P and Q with radii equal to the distance between P and Q. These circles will intersect at two points, which we'll call A and B. Any point outside of these two circles will be closer to either P or Q than the other point.

Now, imagine drawing a third circle centered at the midpoint of segment AB with a radius equal to half the distance between P and Q. This circle will contain all the points that are closer to P than to Q. To see why, imagine drawing a line tangent to this circle at point A. Any point on the same side of this line as P will be closer to P than to Q. Similarly, any point on the opposite side of this line as Q will be closer to P than to Q.

In conclusion, the set of all points in a plane that are closer to P than to Q is the interior of a circle centered at the midpoint of segment PQ with a radius equal to half the distance between P and Q.

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The equation of the line in the graph is:

y = 2

A general linear equation can be written as:

y = ax + b

Where a is the slope and b is the y-intercept.

Particularly, in this case we can see that the line intercepts the y-axis at the value y = 2, then we have that b = 2

y = ax + 2

Now we can see that the line also passes through the point (5, 2), replacing these values in the equation we get:

2 = a*5 + 2

2 - 2 = a*5

0 = a*5

0/5 = a

0 = a

Then the equation of the line is:

y = 2

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The null hypothesis is that the proportion of residents in favor of annexation is equal to or greater than 44%, while the alternative hypothesis is that the proportion is less than 44%. A significance level of 0.02 will be used to determine if there is sufficient evidence to support the mayor's claim.

The null hypothesis (H0) is a statement of no difference or no effect, while the alternative hypothesis (H1) is a statement of the expected difference or effect. In this case, the null hypothesis is that p >= 0.44, where p is the proportion of residents in favor of annexation. The alternative hypothesis is that p < 0.44.

To test the hypothesis, a sample of residents can be taken and the proportion of those in favor of annexation can be calculated. Then, a hypothesis test can be performed using the sample proportion and the null hypothesis to determine if there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. A significance level of 0.02 means that if the p-value (the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true) is less than 0.02, then the null hypothesis will be rejected in favor of the alternative hypothesis.

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Answer: Correct.

Step-by-step explanation:

2.8 Kg is 2800 grams

Each day birds eat 125 grams, 125x7 = 875 grams in a week.

If we divide the 875 grams which is that total feed in a week to 2800 our total feed, we will find the argument is correct or not.

2800/875 = 3,2

So it is correct.

Answer:

4.695

Step-by-step explanation:

multiply top and bottom by 100, 000 to clear any decimal places

now we have 78, 000 / 16,614

= 4.695 (3 decimal places).

Answer:

[tex]\frac{3}{2}[/tex]

Step-by-step explanation:

as x → 1 , the denominator = 2x - 2 = 2(1) - 2 = 2 - 2 = 0

this means the expression is undefined

simplify the expression by factoring numerator and denominator

x³ - 1 ← is a difference of cubes and factors in general as

a³ - b³ = (a - b)(a² + ab + b²) , then

x³ - 1

= x³ - 1³ ( with a = x and b = 1 )

= (x - 1)(x² + x + 1)

2x - 2 = 2(x - 1)

rewriting the expression

lim x → 1 [tex]\frac{(x-1)(x^2+x+1)}{2(x-1)}[/tex] ← cancel (x - 1) on numerator/ denominator

lim x → 1 [tex]\frac{x^2+x+1}{2}[/tex]  ( substitute x = 1 )

limx→ 1 [tex]\frac{1+1+1}{2}[/tex] = [tex]\frac{3}{2}[/tex]

When we evaluate the expression lim x → 1 | x³ - 1 | / (2x - 2), the result obtained is 3/2

First, we shall express x³ - 1 in factor from. This is illustrated below:

x³ - 1 = x³ - 1³ => Difference of cubes

Thus, we have:

x³ - 1 = (x - 1)(x² + x + 1)

Next, we shall express 2x - 2 in factor form. Details below:

2x - 2

2 is common in both terms

Thus,

2x - 2 = 2(x - 1)

Finally, we shall evaluate lim x → 1 |x³ - 1| / (2x - 2). This is shown below:

|x³ - 1| / (2x - 2) = [(x - 1)(x² + x + 1)] / 2(x - 1)

Cancel out (x - 1)

|x³ - 1| / (2x - 2) = (x² + x + 1) / 2

As x tends to 1, we have

|x³ - 1| / (2x - 2) = (1² + 1 + 1) / 2

|x³ - 1| / (2x - 2) = 3 / 2

Thus, we can conclude that the evaluation of lim x→1 |x³ - 1| / (2x - 2) is 3/2

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Answer:

[tex]5 + 2\sqrt{3}[/tex]

    &          

[tex]5-2\sqrt{3}[/tex]

Step-by-step explanation:

Okay! The equation is : 2x²+26=20x

Right off the bat we notice that this can be simplified. We divide all numbers by a common multiple: 2.

Our resulting equation is [tex]x^2 + 13= 10x[/tex]

Now, in order to plug this equation into our quadratic formula, we need to rearrange this equation into the [tex]ax^2 + bx +c = 0[/tex] format.

In order to do that, we simply move the 10x to the left side of the equation, resulting in this: [tex]x^2 - 10x + 13[/tex]

Here is the quadratic formula:

(-b±√(b²-4ac))/ 2a

I will include a picture of the quadratic equation at the bottom (because the typed equation is strange).

So looking at our previously found formula, x^2 - 10x + 13, we know that a: 1

b: -10

c: 13

Now, we plug in our values!

(-(-10) ± √((-10)²-(4(1)(13))) / 2(1)

Simplify! (10 ± √(100-52)) / 2

Simplify again! (10 ± √48) / 2

Now we must simplify the square root. If we try to find the square root of 48, it comes out to 6.92820323, which is a very messy number. We will NOT be using this number. We will instead find the factors of 48.

2·2·2·2·3 = 48

So it looks like this: √2·2·2·2·3

We can pair up the similar numbers, so it looks like: √(2·2)(2·2)·3

Now, we move the pairs of twos to the front of the equation (but only one two from each pair is represented because they've been square-rooted) , and out of the square root, to get us: 2·2 √3, which equals 4√3

Now that we have the square root figured out, we re-enter the square root into the equation we had before (replacing the un-simplified version with the simplified version), which was (10 ± √48) / 2.

Here is the equation with the simplified root:  (10 ± 4√3) / 2

Now we notice that 10 and 4 are divisible by 2, so the equation becomes: (5 ± 2√3), which is 5+2√3, AND 5-2√3

Hope that helped!!!!

Answer:

Step-by-step explanation:

We must divide the frequency of F grades by the total number of pupils in the class to get the likelihood that a student would receive an F.

The frequency distribution being what it is:

Level: A B C D F

28 35 56 14 7 times each year

You can determine the total number of pupils by adding the frequencies:

Students total: 28 + 35 + 56 + 14 + 7 = 140.

We can now determine the probability:

P(F) = Number of students overall / Frequency of F grades

P(F) = 7 / 140 P(F) = 0.05

As a result, the likelihood that a student will receive a F is 0.05, or 5%.

We are given a sample of n=712 20 29 year old men, with a sample mean weight of 183 pounds. We are also told that the population standard deviation is 40 pounds.

To calculate the confidence interval, we use the formula

Zα/2 is the critical value of the standard normal distribution at a significance level of α/2, σ is the population standard deviation, and n is the sample size.

The critical value of the standard normal distribution at a 90% confidence level is 1.645. Plugging in the provided values, we get:

CI = 183 1.645 * (40/√712)

CI = [179.86, 186.14]

This means we can be 90% confident that the true population mean weight of 20 29 year old men lies within the range of 179.86 to 186.14 pounds.

In summary, based on the sample of 712 men and the given population standard deviation of 40 pounds, we can estimate the population mean weight with 90% confidence to be between 179.86 and 186.14 pounds.

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A tax rate of $0.0711 per $1,000 of assessed valuation in decimal form is 0.00711% and in fractional form is $0.0000711 .

Given, that tax rate of $.0711 .

First, divide the tax rate by 1,000 to determine the rate per dollar:

$0.0711 / 1,000 = $0.0000711.

This represents the decimal equivalent of the tax rate per dollar.

To express it as a percentage, multiply the decimal value by 100: $0.0000711 × 100 = 0.00711%.

Therefore, a tax rate of $0.0711 per $1,000 of assessed valuation is equal to 0.00711% in decimal form.

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A tax rate of $.0711 per dollar is equivalent to $71.1 per $1000 when expressed in terms of assessed valuation.

A tax rate of $.0711 in decimal form expresses a tax of 7.11 cents per dollar. However, the question asks for the tax rate expressed per $1000. Therefore, to find this, we need to multiply the tax rate per $1 by 1000. Thus, $.0711 per $1 x 1,000 = $71.1 per $1000 of assessed valuation.

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The value of tan F in this right-angled triangle is approximately 3.42857.

In the right-angled triangle DEF, with a right angle at E, we are given the length of the perpendicular DE, which is 24 cm, and the length of the base EF, which is 7 cm.

To find the value of tan F, we can use the tangent function, which is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.

In this case, F is the angle opposite to side DE, and the adjacent side is EF. So, we can write:

tan F = DE / EF

Substituting the given values, we have:

tan F = 24 cm / 7 cm

Now, we can divide 24 by 7:

tan F ≈ 3.42857

So, the value of tan F in this right-angled triangle is approximately 3.42857.

This means that the ratio of the length of the perpendicular side DE to the length of the base side EF is approximately 3.42857. It indicates how steep or inclined the line EF is with respect to the line DE in the triangle DEF.

Remember that tangent is a trigonometric function that relates the angles of a right triangle to the lengths of its sides. In this case, we used it to find the ratio of the sides in the triangle and determine the value of tan F.

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The value of

1. BC = 12.04

2. angle B = 50°

3. angle C = 50°

Pythagoras theorem states that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse

c² = a² + b²

BC is the hypotenuse

c² = 8² + 9²

c² = 64+81

c² = 145

c = √145

= 12.04

TanB = opp/adj

tanB = 6/8

tanB = 0.75

B = 40( nearest degree)

angle C = 90- 40

C = 50°

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a) H is a subgroup of G. b) H is not a subgroup of G. c) H is a subgroup of G. d) H is a subgroup of G. e) H is not a subgroup of G. f) H is a subgroup of G. g) H is not a subgroup of G. h) H is a subgroup of G.

(a) H = {0, 1, -1}, G = Z:

H is a subgroup of G since it is closed under addition, inverse and contains the identity element.

(b) H = {1, 3}, G = Zs:

H is not a subgroup of G since it is not closed under addition. For example, 1 + 3 = 4 is not in H.

(c) H = {1, 3}, G = Z*_15:

H is a subgroup of G since it is closed under multiplication, inverse and contains the identity element.

(d) H = {element_1 (1, 2) (3 4), (1, 2)(2, 4)}, G = S_3:

H is a subgroup of G since it is closed under composition, inverse and contains the identity element.

(e) H = {[1 0 0 1], [-1 0 0 -1], [0 1 -1 0], [0 -1 1 0]}, G = GL_1(z):

H is not a subgroup of G since it is not closed under matrix multiplication. For example, [1 0 0 1] * [0 1 -1 0] = [0 1 -1 0] is not in H.

(f) H = {2, 4, 6}, G = Z_6:

H is a subgroup of G since it is closed under addition, inverse and contains the identity element.

(g) H = N, G = Z:

H is not a subgroup of G since it does not contain the identity element.

(h) H = {(m, k)|m + k is even}, G = Z x Z:

H is a subgroup of G since it is closed under addition, inverse and contains the identity element.

"Z" refers to the integers and "Z*_15" refers to the integers modulo 15. "GL_1(z)" refers to the set of invertible 1x1 matrices with integer entries.

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State, AAA rated and High tech firm bonds have maturity of 10years, 5years and 3 years respectively.

Bond is a financial instrument that represents a debt obligation. When you purchase a bond, you are essentially lending money to the issuer (which can be a government, corporation, or other entity) in exchange for regular interest payments and the repayment of the principal amount at a specified future date, known as the maturity date.

1. Maturity of Each Bond Group:

- State bonds: The state bonds have a maturity of 10 years.

- AAA rated bonds: The AAA rated bonds have a maturity of 5 years.

- High tech firm bonds: The high tech firm bonds have a maturity of 3 years.

2. Coupon Rates:

- State bonds: The state bonds have a 5% coupon rate.

- AAA rated bonds: The AAA rated bonds have a 6.5% coupon rate.

- High tech firm bonds: The coupon rate is not provided for the high tech firm bonds.

3. Par Values:

- State bonds: The par value of each state bond is not mentioned.

- AAA rated bonds: The par value of each AAA rated bond is not mentioned.

- High tech firm bonds: The par value of each high tech firm bond is $500.

4. Municipal Bonds:

Based on the given information, there is no mention of Sylvester purchasing any municipal bonds. Therefore, none of Sylvester's new investments are municipal bonds.

5. Corporate Bonds:

- High tech firm bonds: The high tech firm bonds are issued by a high tech firm, which indicates that they are corporate bonds.

6. Standard & Poor's Consultation:

From the information provided, there is no specific indication as to which bond purchase Sylvester consulted Standard & Poor's for.

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In diagram A, the axis of symmetry will be vertical and has rotational asymmetry. In diagram D, the axis of symmetry will be horizontal.

Axial symmetrical is similarity around an axis; an item is internally symmetric if it retains its appearance when turned around an axis.

Rotational symmetry is a property of an object or a figure that remains unchanged when it is rotated about a fixed point by an angle less than 360 degrees.

In diagram A, the axis of symmetry will be vertical and has rotational asymmetry.

In diagram D, the axis of symmetry will be horizontal.

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The volume of the sphere is  0. 52 ft³

To determine the volume, we need to know the formula

Hence, the formula that is used for calculating the volume of a sphere is expressed as;

V = 4/3 πr³

Such that the parameters of the formula are;

Substitute the values, we have that;

Volume = 4/3 ×3.14 × 0.5³

Find the cube value

Volume = 4/3 × 3.14 × 0.125

Multiply the values

Volume = 1.57/3

Divide the values

Volume = 0. 52 ft³

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Answer:

8.875 km

Step-by-step explanation:

1km = 1000 metres

so 8875 metres = 8875/1000 = 8.875 X 1km = 8.875km

Answer: 8,875 metres= 8.875 kilometres

The required value of the account after 2 years is $337.08.

The equation below represents the value V of a bank account in which $300 is invested at 6.00% interest compounded yearly:

[tex]V = 300 \times (1.06)^t[/tex]    .....(i)

where t is the period in years.

It is required to find the value of the account after 2 years.

The value of the bank account after 2 years can be found by substituting t = 2 into the given equation (i):

V = 300 × (1.06)²

V = 300 × 1.1236

Apply the multiplication operation to get

V = 337.08

Therefore, the value of the account after 2 years is $337.08.

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The rule for the translated function is:

g(x) = log(x) + 2

We know that the function f(x) is the parent logarithmic function, it can be written as.

f(x) = log(x)

We know that g(x) is a translation of f(x), and we can see that the graph of g(x) is 2 units above the graph of f(x), then we can write:

g(x) = f(x) + 2

Now we can replace the function f(x) there to get:

g(x) = log(x) + 2

That is the translated function.

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Complete question:

"Which of the following functions describes g?

g(x) = log(x) + 2

g(x) = log(x + 2)

g(x) = log(x) - 2"

Answer:

B is 41 × 1/2

Step-by-step explanation:

A is 18 × 1/2 so that is your answer

The area of a triangle with sides A=18, B=41, and C=41 is 360cm^2(assuming sides are given in "cm").

By using Heron's formula to calculate the area of a triangle from the sides,

Just use this two-step process:

Step 1: Calculate "s" (half of the perimeter of the triangle):

s =  a+b+c/2

Step 2: Then calculate the Area:

A = \sqrt{s(s−a)(s−b)(s−c)}

Here, a=18,b=41,c=41

so,s=50 and A=360.

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The set of all two-letter strings can be thought of as an ordered pair of two letters, where each letter can be selected from the alphabet {a, b, ..., z}.

Since there are 26 letters in the alphabet, there are 26 choices for the first letter in the string. For the second letter, however, there are only 25 choices, since we cannot repeat the letter selected for the first position. Thus, the number of different two-letter strings is the product of the number of choices for each letter, which is 26 * 25 = 650.

This problem illustrates the concept of counting principles, specifically the product rule of counting. The product rule states that the total number of outcomes for a sequence of events is the product of the number of outcomes for each event. In this case, the two events are the selection of the first letter and the selection of the second letter. By applying the product rule, we can easily determine the total number of possible two-letter strings.

This type of problem is commonly encountered in combinatorics, which is the branch of mathematics concerned with counting and arranging objects. The ability to count and calculate the number of possible outcomes is important in many fields, including probability theory, statistics, and computer science.

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The scenario that the sample means differ more is Scenario 1 because the mean seems to be very far from each other.

The scenario that there is a larger variation in the distribution of data within each sample is Secnario 2

The f statistics is larger for Scenario 1 as the Means are far and the spread is less so the F statistic will be larger.

Scenario 1 as F statistic will be larger. So chances of Rejecting H0 is more.

The F statistic is a statistical measure used to determine if there is a significant difference between the means of two or more groups. It is calculated by dividing the between-group variability (also known as the mean square between) by the within-group variability (also known as the mean square error).

The scenario that there is a larger variation in the distribution of data within each sample is Secnario 2 because for this scenario there seems to be more spread within each sample.

Lastly,  Scenario 1 as F statistic will be larger. So chances of Rejecting H0 is more.

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The answer is A) i.e. Both will increase.

If the point in the upper left corner of the Explanatory variable versus the Response variable scatterplot is removed, the correlation (r) and the slope of the line of best fit (b) will change.

In a data graph or dataset you're dealing with, an outlier is a data point that is unusually high or unusually low in comparison to the closest data point and the rest of the nearby coexisting values. if the point is an outlier, removing it will increase the correlation (r) and the slope of the line of best fit (b).

Therefore, the answer is A) Both will increase.

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Given question is incomplete, the complete question is below

If the point in the upper left corner of the Explanatory variable versus the Response variable scatterplot shown below is removed, what will happen to the correlation (r) and the slope of the line of best fit (b) ?

Possible answers

A) Both will increase

B) r will increase and b will decrease

C) They will not change

D) r will decrease and b will increase

E) Both will decrease

The number of ways to select 6 pieces of candy from a bag containing 14 pieces can be calculated using the combination formula. There are 3003 different ways to select six pieces of candy from a bag containing 14 pieces.

which is:

nCr = n! / (r! * (n-r)!)

where n is the total number of candies in the bag (14), and r is the number of candies to be selected (6).

Using this formula, we get:

14C6 = 14! / (6! * (14-6)!)
= 3003

Therefore, there are 3003 ways to select 6 pieces of candy from a bag containing 14 pieces.

To determine the number of ways six pieces can be selected from a bag containing 14 pieces of candy, you can use the concept of combinations. Combinations are used when the order of selection does not matter.

The formula for combinations is:
C(n, r) = n! / (r!(n-r)!)

In this case, n = 14 (total pieces of candy) and r = 6 (number of pieces to be selected).

Using the formula:

C(14, 6) = 14! / (6!(14-6)!)

C(14, 6) = 14! / (6! * 8!)

Now, calculate the factorials:

14! = 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
6! = 6 * 5 * 4 * 3 * 2 * 1
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

Then, substitute the factorials back into the equation:

C(14, 6) = (14!)/(6! * 8!)

C(14, 6) = (87178291200)/(720 * 40320)

Finally, divide the numerator by the denominator:

C(14, 6) = 3003

So, there are 3003 different ways to select six pieces of candy from a bag containing 14 pieces.

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The range of the graphed exponential function is b . y < -1.

The function which is best represented by the graph is f(x) = -x² + 1.

1) Given an exponential function.

We have to find the range of the function.

The range of the function is the set of all the y values for the x values where the function is defined.

From the graph, it is clear that for any x values, the y values are all either -1 or numbers less than -1.

So the range is y < -1.

2) Given a graph of a parabola opens downwards.

So the function will be quadratic. That is, the highest degree of the variable will be 2.

For a function of the form, (parent function), y = -x², the parabola passes through the point (0, 0), which will be the vertex and the parabola is opened downwards.

Here vertex is (0, 1).

That is the parabola is shifted up to 1 unit.

A function f(x) after the translation to k units up becomes f(x) + d.

So here since the original function is shifted up 1 units, it becomes,

f(x) = -x² + 1

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Tanisha would expect to earn 180 on a day when she waits on 10 tables.

Tanisha's guaranteed wage is 40 per day. This means that no matter what happens during her shift, she will earn at least 40.

In addition to her guaranteed wage, Tanisha earns money from tips. She earns about 14 in tips for each table she waits on. If Tanisha waits on 10 tables in a day, she can expect to earn:

14 x 10 = 140

Therefore, her total earnings for the day would be:

40 (guaranteed wage) + 140 (tips) = 180

So Tanisha would expect to earn 180 on a day when she waits on 10 tables.

It's important to note that Tanisha's tips may vary from day to day depending on factors such as the size of the party, the generosity of the customers, and the overall volume of business at the restaurant. However, based on past experience, she can reasonably expect to earn around 14 in tips for each table she waits on.

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To prove the equation [tex]\frac{1}{\tan(2\theta) - \tan(\theta)} - \frac{1}{\cot(2\theta) - \cot(\theta)} = \cot(\theta)[/tex] we'll simplify the left side, this is shown below:

Using the trigonometric identities [tex]\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}[/tex] and [tex]\cot(\theta) = \frac{1}{\tan(\theta)}[/tex]

We can rewrite the expression as [tex]\frac{1}{\tan(\theta)(1 - \tan(\theta))} - \frac{\tan(\theta)}{\tan(\theta)(1 - \tan^2(\theta))}[/tex]

Combining the fractions with a common denominator, we obtain [tex]\frac{1 - \tan(\theta)}{\tan(\theta)(1 - \tan(\theta))}[/tex]

Simplifying further, we cancel out the [tex](1 - tan(\theta))[/tex] terms, leaving us with [tex]\frac{1}{\tan(\theta)}[/tex] = [tex]\cot(\theta)[/tex], which is equivalent to the right side of the equation.

Thus, the equation is proven.

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