Maggie has a circular table cloth with a 72-inch diameter that she plans to sew lace around. if the lace comes in 3-foot rolls, how many rolls will she need?

area = sum of parallel sides . height/2

= (5.9 + 16.3) 4.6/2

= 22.2 . 2.3

= 51.06 m²

so the answer is A

Question:

In the right triangle shown, ∠B=60° and BC = 2√3

How long is AB?

Answer exactly, using a radical if needed.

The image of the triangle is attached below:

Answer:

The length of AB is [tex]4\sqrt{3}[/tex]

Explanation:

It is given that ∠B = 60° and BC = [tex]2\sqrt{3}[/tex]

To determine the length of AB, we shall use the cosine formula.

Because the value of the angle and its adjacent side is given and AB is the hypotenuse, we shall substitute the value of angle and adjacent side in the formula to find the value of AB.

Thus, the formula for [tex]\cos \theta[/tex] is given by

[tex]\cos \theta=\frac{a d j}{h y p}[/tex]

Where [tex]\theta=60[/tex] and [tex]adj= 2\sqrt{3}[/tex] and [tex]hyp=x[/tex]

Substituting these values in the formula, we get,

[tex]\cos 60=\frac{2 \sqrt{3}}{x}[/tex]

Interchanging, we get,

[tex]x=\frac{2 \sqrt{3}}{\cos 60}[/tex]

The value of [tex]cos 60 =\frac{1}{2}[/tex]

Substituting, we get,

[tex]x=\frac{2 \sqrt{3}}{\frac{1}{2} }[/tex]

[tex]x=4\sqrt{3}[/tex]

Thus, the value of x is [tex]4\sqrt{3}[/tex]

Hence, the length of the hypotenuse AB is [tex]4\sqrt{3}[/tex]

Answer: [tex]m=-21[/tex]

Step-by-step explanation:

In order to solve the exercise, you need to remember the following properties:

1. Addition property of equality:

If [tex]a=b[/tex], then [tex]a+c=b+c[/tex]

2. Subtraction property of equality:

If [tex]a=b[/tex], then [tex]a-c=b-c[/tex]

3. Divison property of equality:

If [tex]a=b[/tex], then [tex]\frac{a}{c}=\frac{b}{c}[/tex]

4. Multiplication property of equality:

If [tex]a=b[/tex], then [tex]a*c=b*c[/tex]

Then, given the following equation:

[tex]9+\frac{m}{3}=2[/tex]

You need to followw these steps in order solve for "m":

- Apply the Subtraction property of equality subtracting 9 from both sides of the equation:

[tex]9+\frac{m}{3}-9=2-9\\\\\frac{m}{3}=-7[/tex]

- Apply the Multiplication property of equality multiplying both sides of the equation by 3. Then, you get:

[tex](3)(\frac{m}{3})=(-7)(3)\\\\m=-21[/tex]

The calculation of take home pay is the actual amount calculation of how much you will be credited on a job done well.

Step-by-step explanation:

Before calculation, things to be taken care of are -

Now knowing these much in exact form of amount, we may proceed to calculation -

The area of flying disk is 452.16 square centimeters

Step-by-step explanation:

The disc is in circular shape so we will use the formulas for circle in this question.

Given

Circumference = C = 75.36 centimeters

We have to find the area of the disc for which we have to find the radius first.

Let r be the radius

Then

Circumference is given by:

[tex]C=2\pi r[/tex]

putting the values

[tex]75.36 = 2*3.14*r\\75.36 = 6.28r[/tex]

Dividing both sides by 6.28

[tex]\frac{6.28r}{6.28} = \frac{75.36}{6.28}\\r = 12\ cm[/tex]

The area of circle is given by:

[tex]A = \pi r^2[/tex]

Putting the values

[tex]A = 3.14 * (12)^2\\A = 3.14*144\\A =452.16\ cm^2[/tex]

Hence,

The area of flying disk is 452.16 square centimeters

Keywords: Circle, area

Learn more about circle at:

#LearnwithBrainly

Answer:

A (5x + 3)(5x – 3)

B (7x + 4)(7x + 4)

E (x – 9)(x – 9)  

F (–3x – 6)(–3x + 6)

in other words, A,B,E, and F on edge-nuity. just completed with 100%

Step-by-step explanation:

Answer:

Step by-step explanation:

Speed v = 12km/h = 12/60km/min

Time t = 75min =

Distance d = ?

V = d/t

12/60 = d/75

Cross multiply

60d = 75 × 12

60d = 900

d = 900/60

d = 15km

f(x) = 2([tex]2^{x}[/tex])

Step-by-step explanation:

Step 1 :

Given

When  x =0, f(x) = 2

x =1, f(x) = 4

x =2, f(x) = 8

x =3, f(x) = 16

Step 2 :

Substituting for x in the function f(x) = 2([tex]2^{x}[/tex]) we get,

When  x =0, f(x) = 2

x =1, f(x) = 4

x =2, f(x) = 8

x =3, f(x) = 16

This matches the given sequence and this shows that the function represents the given sequence .

Final answer:

The function rule for the given sequence 2, 4, 8, 16, ... is [tex]f(x) = 2^x[/tex], representing the sequence's exponential growth pattern where each term is double the previous term.

Explanation:

The given sequence is 2, 4, 8, 16, ..., which is a geometric sequence where each term is multiplied by 2 to get the next term. This pattern corresponds to an exponential function where the base is 2 and the exponent is the position of the term, which can also be thought of as the function's input, x. Therefore, the function that represents this sequence is f(x) = 2x.

The other function choices given, such as f(x) = 2x + 2 or f(x) = 2, do not correctly represent the pattern observed in the sequence. The correct function rule for the given sequence is f(x) = 2x which fits the pattern where when x is substituted for each term's position (starting from 0), the function provides the corresponding term of the sequence.

Answer:D

Step-by-step explanation:

Multiply the cost of the house by 0.18.

Following the slope of the line at x= 0 the line is at y =1 , at x=, it is at Y = 1.5, so the slope is 1/2

The y intercept is where the line crosses the y a is at x =0 which is 1

The equation would be y = 1/2x+ 1

Answer:

y = 1/2x+ 1

Step-by-step explanation:

63 divided by 9 is 7, so -7+7 is 0

Hope this helped

Answer:

0

Step-by-step explanation:

using PEMDAS

63/9 = 7

-7 + 7 = 0

Answer:

x=-1/3, y=-13/3. (-1/3, -13/3).

Step-by-step explanation:

y=x-4

4x-y=3

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

4x-(x-4)=3

4x-x+4=3

3x+4=3

3x=3-4

3x=-1

x=-1/3

y=-1/3-4

y=-1/3-12/3=-13/3

Answer: ([tex]-\frac{1}{3},-\frac{13}{3}[/tex]

Step-by-step explanation:

Substitut (x-4) into the equation of 4x-y=3

(Substitute what y equals into the equation

Make sure to keep parentheses!

That becomes 4x-(x-4)=3

Than you must distribute the negative to x and to -4

When you do it creates the equation of 4x-x+4=3

When combining like terms you get: 3x+4=3

Then solve                                                -4   -4

3x=-1

3    3               x=-1    

                           3

Than substitute the x in for the equation of y=x-4 to find what y equals! since you know that x equals -1/3 than subtract -1/3 - 4 to get

y= -13          

     3        

Answer:

x= 1÷4

Step-by-step explanation:

2x - 6x + 1 =0

-4x + 1= 0

-4x = -1

Final answer:

Using the quadratic formula, the real solutions of the equation [tex]x^2 - 6x + 1 = 0[/tex]are x = 3 + 2√2 and x = 3 − 2√2 since the discriminant is positive, indicating two real solutions.

Explanation:

To find all real solutions of the quadratic equation  [tex]x^2 - 6x + 1 = 0[/tex], we can use the quadratic formula, which is x = −b ± √(b2 − 4ac) / (2a) where a, b, and c are coefficients from the equation in the form [tex]ax^2 + bx + c = 0[/tex]. For our equation, a = 1, b = −6, and c = 1. Let's apply these values into the formula:

Calculate the discriminant: (−6)2 − 4(1)(1) = 36 − 4 = 32.

Since the discriminant is positive, there are two real solutions.

Calculate the solutions: x = (6 ± √32) / (2 × 1) = (6 ± 4√2) / 2 = 3 ± 2√2.

The real solutions of the equation are x = 3 + 2√2 and x = 3 − 2√2.

Answer:

  $2.40

Step-by-step explanation:

You have ...

  finance charge = 1.2% × unpaid balance

     = 0.012 × $200 . . . . . fill in the value of unpaid balance

  finance charge = $2.40

Answer:

Proof in the explanation

Step-by-step explanation:

Trigonometric Equalities

Those are expressions involving trigonometric functions which must be proven, generally working on only one side of the equality

For this particular equality, we'll use the following equation

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

The equality we want to prove is  

[tex]\displaystyle arccos\ \sqrt{\frac{2}{3}}-arccos\left(\frac{1+\sqrt{6}}{2\sqrt{3}}\right)=\frac{\pi}{6}[/tex]  

Let's set the following variables:

[tex]\displaystyle x=arccos\ \sqrt{\frac{2}{3}},\ y=arccos(\frac{1+\sqrt{6}}{2\sqrt{3}})[/tex]

And modify the first variable:

[tex]\displaystyle x=arccos\ \frac{\sqrt{6}}{3}}=>\ cos\ x= \frac{\sqrt{6}}{3}}[/tex]

Now with the second variable

[tex]\displaystyle y=arccos\ \frac{1+\sqrt{6}}{2\sqrt{3}}=>cos\ y=\frac{1+\sqrt{6}}{2\sqrt{3}}=\frac{\sqrt{3}+3\sqrt{2}}{6}[/tex]

Knowing that

[tex]sin^2x+cos^2x=1[/tex]

We compute the other two trigonometric functions of X and Y

[tex]\displaystyle sin \ x=\sqrt{1-cos^2\ x}=\sqrt{1-(\frac{\sqrt{6}}{3})^2}=\sqrt{1-\frac{6}{9}}=\frac{\sqrt{3}}{3}[/tex]

[tex]\displaystyle sin\ y=\sqrt{1-cos^2y}=\sqrt{1-\frac{(\sqrt{3}+3\sqrt{2})^2}{36}}}[/tex]

[tex]\displaystyle sin\ y=\sqrt{\frac{36-(3+6\sqrt{6}+18)}{36}}=\sqrt{\frac{15-6\sqrt{6}}{36}}[/tex]

Computing

[tex]15-6\sqrt{6}=(3-\sqrt{6})^2[/tex]

Then

[tex]\displaystyle sin\ y=\frac{3-\sqrt{6}}{6}[/tex]

Now we replace all in the first equality:

[tex]\displaystyle cos(x-y)=\frac{\sqrt{6}}{3}.\frac{\sqrt{3}+3\sqrt{2}}{6}+\frac{\sqrt{3}}{3}.\frac{3-\sqrt{6}}{6}[/tex]

[tex]\displaystyle cos(x-y)=\frac{3\sqrt{2}+6\sqrt{3}}{18}+\frac{3\sqrt{3}-3\sqrt{2}}{18}[/tex]

[tex]\displaystyle cos(x-y)=\frac{9\sqrt{3}}{18}=\frac{\sqrt{3}}{2}=cos\ \pi/6[/tex]

Thus, proven  

[tex]\text{Hey there!}[/tex]

[tex]\bf{What\ is\ greater: \dfrac{3}{10}, \dfrac{1}{2},\&\dfrac{2}{5}?}[/tex]

[tex]\bf{Well, let's\ turn\ our\ fractions\ into\ decimals\downarrow}[/tex]

[tex]\bullet\ \bf{\dfrac{3}{10}=0.30(also\ known\ as\ 0.3)}\\\\\bullet \ \bf{\dfrac{1}{2}=0.50(also\ known\ as\ 0.5)}\\\\\bullet \ \bf{\dfrac{2}{5}=0.40(also\ known\ as\ 0.4)}[/tex]

[tex]\bf{The\ fractions\ in\ percentages\downarrow}[/tex]

[tex]\bullet \ \mathsf{\dfrac{3}{10}=30\%}\\\\\bullet \ \mathsf{\dfrac{1}{2}=50\%}\\\\\bullet \ \mathsf{\dfrac{2}{5}=40\%}[/tex]

[tex]\text{Now that we have that out the way. Let's solve for your answer!}[/tex]

[tex]\bf{\dfrac{1}{2}}\rightarrow\dfrac{2}{5}\rightarrow\dfrac{3}{10}}}} \leftarrow( that's\ for\ GREATEST\ to\ LEAST)[/tex]

[tex]\boxed{{\bf{Answer: \dfrac{1}{2}\ would\ be\ your\ GREATEST\ or \ BIGGEST\ one}}}\checkmark[/tex]

[tex]\text{Good luck on your assignment and enjoy your day!}[/tex]

~[tex]\frak{LoveYourselfFirst:)}[/tex]

Answer:

[tex]f(x) = (840)\times (1.05)^{x}[/tex] where f(x) is the bonus every year and x is in number of years

Step-by-step explanation:

The function that represents the situation where $840 annual bonus increases by 5% each year is given by

[tex]f(x) = (840)\times (1.05)^{x}[/tex] where f(x) is the bonus every year and x is in number of years

Final answer:

The function representing an $840 annual bonus that increases by 5% each year is f(x) = 840(1 + 0.05)ˣ

Explanation:

To write a function that represents the situation where an $840 annual bonus increases by 5% each year, we can use an exponential growth model.

The general form of an exponential growth function is f(x) = a(1 + r)ˣ, where a is the initial amount, r is the growth rate, and x is the number of time periods.

In this case, the initial bonus a is $840, the growth rate r is 5% or 0.05, and x represents the number of years. Thus, the function can be written as:

f(x) = 840(1 + 0.05)ˣ

Answer:

  4 and 5

Step-by-step explanation:

  20 is greater than 4² = 16, and less than 5² = 25. So, the square root of 20 is between 4 and 5.

___

Your calculator will tell you that √20 ≈ 4.472, so is between 4 and 5.

Answer:

y = 38°

Step-by-step explanation:

If Angle x and angle y are complementary, then x + y = 90°

Also, if angle x is supplementary to a 128 angle, then x + 128° = 180°.

Solving the latter equation for x, we get x = 52°.

Therefore, by the first equation, x + y = 90° becomes 52° + y = 90°, yielding

y = 38°

Answer:

[tex]\boxed{\mathtt{x=52^{\circ}}}[/tex]

[tex]\boxed{\mathtt{y=38^{\circ}}}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to solve for the measures of angles x and y.}[/tex]

[tex]\textsf{First, x is supplementary to an angle that equals 128}^{\circ}.[/tex]

[tex]\Large\underline{\textsf{What are Supplementary Angles?}}[/tex]

[tex]\textsf{Supplementary angles are 2 or more angles that add up to \underline{180}}^{\circ}.[/tex]

[tex]\large\underline{\textsf{This means that;}}[/tex]

[tex]\mathtt{x+128^{\circ}=180^{\circ}.}[/tex]

[tex]\large\underline{\textsf{Subtract 128 from both sides for x:}}[/tex]

[tex]\boxed{\mathtt{x=52^{\circ}}}[/tex]

[tex]\Large\underline{\textsf{What are Complementary Angles?}}[/tex]

[tex]\textsf{Complementary angles are 2 or more angles that add up to \underline{90}}^{\circ}.[/tex]

[tex]\large\underline{\textsf{This means that;}}[/tex]

[tex]\mathtt{x+y=90^{\circ}.}[/tex]

[tex]\textsf{Because we know x, we can solve for y.}[/tex]

[tex]\large\underline{\textsf{Substitute:}}[/tex]

[tex]\mathtt{52^{\circ}+y=90^{\circ}.}[/tex]

[tex]\large\underline{\textsf{Subtract 52 from both sides for y:}}[/tex]

[tex]\boxed{\mathtt{y=38^{\circ}}}[/tex]

Answer:

5 + 2[tex]\sqrt{6}[/tex]

Step-by-step explanation:

Multiply top and bottom by sqrt(3) + sqrt(2)

Top = (sqrt (3) + sqrt(2)) * (sqrt(3) + sqrt(2))

Top = 3 + sqrt(6) + sqrt(6) + 2

Top = 5 + 2sqrt(6)

Bottom = (sqrt (3) - sqrt(2)) * (sqrt(3) + sqrt(2))

Bottom = (3 - sqrt(6) + sqrt(6) - 2)

Bottom = 1

So, the answer is 5 + 2[tex]\sqrt{6\\}[/tex]

Answer: 5+2√6

Step-by-step explanation: attachment

Answer:

n = 5

Step-by-step explanation:

Step  1  :

Step  2  :

Pulling out like terms :

2.1     Pull out like factors :

  -4n + 30  =   -2 • (2n - 15)

Equation at the end of step  2  :

 (0 -  -6 • (2n - 15)) -  -30  = 0

Step  3  :

Step  4  :

Pulling out like terms :

4.1     Pull out like factors :

  12n - 60  =   12 • (n - 5)

Equation at the end of step  4  :

 12 • (n - 5)  = 0

Step  5  :

Equations which are never true :

5.1      Solve :    12   =  0

This equation has no solution.

A a non-zero constant never equals zero.

Solving a Single Variable Equation :

5.2      Solve  :    n-5 = 0

Add  5  to both sides of the equation :

                     n = 5

One solution was found :

                  n = 5

Processing ends successfully

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To solve this, you need to isolate/get the variable "n" by itself in the equation:

-3(-4n + 30) = -30        Divide -3 on both sides

[tex]\frac{-3(-4n+30)}{-3} =\frac{-30}{-3}[/tex]     [two negative signs cancel each other out and become positive]

-4n + 30 = 10       Subtract 30 on both sides

-4n + 30 - 30 = 10 - 30

-4n = -20          Divide -4 on both sides to get "n" by itself

[tex]\frac{-4n}{-4}=\frac{-20}{-4}[/tex]

n = 5

PROOF

-3(-4n + 30) = -30       Substitute/plug in 5 into "n"

-3(-4(5) + 30) = -30

-3(-20 + 30) = -30      Simplify what's inside the parentheses  [PEMDAS}

-3(10) = -30

-30 = -30

Answer:

364 inches cubed

Step-by-step explanation:

Answer:

364 inches cubed

Step-by-step explanation:

Ⓗⓘ ⓣⓗⓔⓡⓔ

Well, the formula for volume is L*H*W

L=4 in, H=13 in, W=7 in

4*13*7=364

(っ◔◡◔)っ ♥ Hope this helped! Have a great day! :) ♥

The value of expression when n = 3 is 22

Solution:

Given expression is:

[tex]\frac{6(n^2 + 2)}{n}[/tex]

We have to find the value of expression when n = 3

Substitute n = 3 in given expression

[tex]\rightarrow \frac{6(3^2 + 2)}{3}\\\\Simplify\ the\ above\ expression\\\\\rightarrow \frac{6(9+2)}{3}\\\\Solve\ the\ terms\ inside\ the\ bracket\\\\\rightarrow \frac{6(11)}{3}\\\\Divide\ 6\ by\ 3 \\\\\rightarrow 2 \times 11 = 22[/tex]

Thus the value of expression when n = 3 is 22

Answer:

22

Step-by-step explanation:

Answer:

45.7%

Step-by-step explanation:

16/14*40%=45.7%

The correct statement is that the profit is 60% if it had been sold for $16.

The amount of something is expressed as if it is a part of the total which is a hundred. The ratio can be expressed as a fraction of 100. The word percent means per 100. It is represented by the symbol ‘%’.Triangle?

Given

When an article was sold for $14 the profit was 40%.

When an article was sold for $14 the profit was 40%.

Then, the cost of the article will be $10.

If the article is sold for $16.

Then profit will be

[tex]\rm Profit = \dfrac{selling\ price \ - \ cost\ price}{cost\ price}*100\\\\\rm Profit = \dfrac{16-10}{10}*100\\\\\rm Profit = 60 \%[/tex]

Thus, the profit will be 60%.

More about the percentage link is given below.

https://brainly.com/question/8011401

The result of the given expression 12 5/6x - 14x + 1/6x  is -x

Given the expression

12 5/6x - 14x + 1/6x

Convert the mixed fractions into improper fractions as shown:

[tex]\frac{77}{6}x - 14x + \frac{1}{6}x[/tex]

Collect the like terms;

[tex]= \frac{77}{6} x+\frac{1}{6}x - 14x\\= \frac{77x+x}{6} - 14x\\= \frac{78x}{6} - 14x\\= 13x - 14x\\= -x[/tex]

Hence the result of the given expression is -x

Learn more here: https://brainly.com/question/17788156

The simplified form of the expression is -x .

Given, expression [tex]12\frac{5}{6}x -14x + \frac{1}{6} x[/tex] .

Expression:  [tex]12\frac{5}{6}x -14x + \frac{1}{6} x[/tex]

Firstly solve the mixed fraction.

[tex]12\frac{5}{6} x\\= \frac{77}{6}x[/tex]

Rewriting the expression :

= [tex]\frac{77}{6}x - 14x + \frac{1}{6}x[/tex]

Solve the terms with common denominator.

= [tex]\frac{77+1}{6} x[/tex]

= [tex]13x[/tex]

Solve the like terms,

[tex]= 13x - 14x\\\\=- x[/tex]

Hence the simplified form of expression is -x .

Know more about expressions,

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Answer:

y=-3/4x+3

Step-by-step explanation:

3x+4y=12

4y=12-3x

4y=-3x+12

y=-3/4x+12/4

y=-3/4x+3

Answer:

The quotient is 345.5

Step-by-step explanation:

Check image

Final answer:

To find the quotient of 1,382 divided by 4, divide each place value separately. The quotient is 343 with a remainder of 2.

Explanation:

To find the quotient of 1,382 divided by 4, we divide the thousands, hundreds, tens, and ones place separately. Starting from the leftmost digit, we divide 1,382 by 4.

4 divided by 1 is 0 with a remainder of 4. Bring down the next digit 3. So, we have 43. Next, divide 43 by 4. 4 divided by 43 is 10 with a remainder of 3. Bring down the next digit 8. So, we have 38. Finally, divide 38 by 4. 4 divided by 38 is 9 with a remainder of 2.

Therefore, the quotient of 1,382 divided by 4 is 343 remainder 2.

To complete the calculations, more data is needed. We only have the angle of elevation of the mother bird, we need one relevant data, as for example, the distance from Lisa to the tree. We'll assume it to be 20 feet.

Answer:

The mother bird is 47 feet above the ground

Step-by-step explanation:

Right Triangles

They are a special type of triangles that have an internal angle of 90°. In such conditions, the following trigonometric relationship is valid:

[tex]\displaystyle tan\theta=\frac{y}{x}[/tex]

Where [tex]\theta[/tex] is the angle of elevation, y is the height of the triangle and x is the horizontal distance to the vertical leg

We can easily find the height of the tree by solving for y

[tex]y=xtan\theta[/tex]

[tex]y=20tan67^o[/tex]

[tex]y=47\ feet[/tex]

The mother bird is 47 feet above the ground

Answer:

(identity has been verified)

Step-by-step explanation:

Verify the following identity:

tan(θ) (cot(θ)^2 + 1)/(tan(θ)^2 + 1) = cot(θ)

Multiply both sides by tan(θ)^2 + 1:

tan(θ) (cot(θ)^2 + 1) = ^?cot(θ) (tan(θ)^2 + 1)

(cot(θ)^2 + 1) tan(θ) = tan(θ) + cot(θ)^2 tan(θ):

tan(θ) + cot(θ)^2 tan(θ) = ^?cot(θ) (tan(θ)^2 + 1)

cot(θ) (tan(θ)^2 + 1) = cot(θ) + cot(θ) tan(θ)^2:

tan(θ) + cot(θ)^2 tan(θ) = ^?cot(θ) + cot(θ) tan(θ)^2

Write cotangent as cosine/sine and tangent as sine/cosine:

sin(θ)/cos(θ) + sin(θ)/cos(θ) (cos(θ)/sin(θ))^2 = ^?cos(θ)/sin(θ) + cos(θ)/sin(θ) (sin(θ)/cos(θ))^2

(sin(θ)/cos(θ)) + (cos(θ)/sin(θ))^2 (sin(θ)/cos(θ)) = cos(θ)/sin(θ) + sin(θ)/cos(θ):

cos(θ)/sin(θ) + sin(θ)/cos(θ) = ^?(cos(θ)/sin(θ)) + (cos(θ)/sin(θ)) (sin(θ)/cos(θ))^2

(cos(θ)/sin(θ)) + (cos(θ)/sin(θ)) (sin(θ)/cos(θ))^2 = cos(θ)/sin(θ) + sin(θ)/cos(θ):

cos(θ)/sin(θ) + sin(θ)/cos(θ) = ^?cos(θ)/sin(θ) + sin(θ)/cos(θ)

The left hand side and right hand side are identical:

Answer: (identity has been verified)

By using the Pythagorean trigonometric identities and substituting the expressions of tan θ, sec θ, and csc θ, we can simplify the given expression to prove that tan θ (1 + cot2 θ) / (1 + tan2 θ) equals cot θ.

To prove that tan θ ( 1 + cot2 θ ) / ( 1 + tan2 θ ) = cot θ, we can use trigonometric identities. Recall the Pythagorean identity which states that cot2 θ + 1 = csc2 θ and tan2 θ + 1 = sec2 θ. Using these identities, we can rewrite the expression on the left side of the equation:

tan θ ( 1 + cot2 θ ) / ( 1 + tan2 θ ) = tan θ * csc2 θ / sec2 θ

Since sec θ = 1/cos θ and csc θ = 1/sin θ, and remembering that tan θ = sin θ / cos θ, we substitute these into the expression:

tan θ * csc2 θ / sec2 θ = (sin θ / cos θ) * (1/sin2 θ) / (1/cos2 θ)

With simplification, the sin2 θ in the numerator and denominator cancel out, as do the cos2 θ terms, leaving us with:

cos θ / sin θ = cot θ

Thus, the original expression simplifies to cot θ.

Answer:

Slope, m = [tex]$ \frac{\textbf{-2}}{\textbf{3}} $[/tex].

Step-by-step explanation:

The slope of the line can be determined by simplifying the given equation to its standard form.

In the standard form, y =  mx + c, m represents the slope of the line.

We are given: [tex]$ y - 2 = - \frac{2}{3}(x + 1) $[/tex]

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

[tex]$ \implies y = - \frac{2}{3}x + \frac{(-2 + 6)}{3} $[/tex]

[tex]$ \implies y = -\frac{2}{3}x + \frac{4}{3} $[/tex]

Comparing it with the standard equation, we can see that the slope, m = [tex]$ \frac{\textbf{-2}}{\textbf{3}} $[/tex].

Hence, the answer.