Help with math problems

Answer:

Step-by-step explanation:

You want the 5th term and the general term of an geometric sequence with a1 = 7 and r = -3.

The general term of an arithmetic sequence is ...

  an = a1·r^(n-1)

For a1=7 and r=-3, the general term is ...

  an = 7·(-3)^(n-1)

Using n=5, the above equation evaluates to ...

  a5 = 7·(-3)^(5 -1)

  a5 = 7·(-3)^4 = 7·81

  a5 = 567

Answer:

2292.03

Step-by-step explanation:

Start with the formula for continuously compounded interest.

Then substitute all given values in the formula.

Finally, solve for the only variable remaining.

[tex] A = Pe^{rt} [/tex]

A = future value = $5000

P = principal (deposited amount) = unknown

r = 6.5% = 0.065

t = time = 12 years

[tex] 5000 = Pe^{0.065 \times 12} [/tex]

[tex] 5000 = Pe^{0.78} [/tex]

[tex]5000 = P \times 2.18147[/tex]

[tex] P = \dfrac{5000}{2.18147} [/tex]

[tex] P = 2292.03 [/tex]

Answer: $2292.03

Answer:

P = $2366.91

(maybe try answering without a comma)

Step-by-step explanation:

The formula for continuous compounding is:

A = Pe^(rt)

Where:

A = final amount

P = principal amount (initial deposit)

e = Euler's number (approximately 2.71828)

r = annual interest rate (as a decimal)

t = time (in years)

We are given:

r = 6.5% = 0.065 (annual interest rate)

t = 12 years

A = $5000 (final amount)

So we can rearrange the formula to solve for P:

P = A / e^(rt)

Substituting the values:

P = 5000 / e^(0.065*12)

P = $2366.91 (rounded to the nearest cent)

Therefore, you would need to deposit $2366.91 in an account with a 6.5% interest rate, compounded continuously, to have $5000 in your account 12 years later.

As, a1/a2 = b1/b2 = c1/c2 = 2 .Thus, the given system of equations forms the identical lines.

A list of numbers x, y, z, etc. that simultaneously make all of the equations true is the solution of a system of equations. A system of equations' solution set is the whole of all possible answers. Finding all answers using formulas containing a certain number of parameters is known as solving the system.

For the given system of equations:

6x - y - 2 = 0

here, coefficients a1 = 6, b1 = -1 and constant c1 = -2

Now,

3x - 1/2y - 1 = 0

here, coefficients  a2 = 3, b2 = -1/2 and constant c2 = -1

taking the ratios:

a1/a2 = 6/3 = 2

b1/b2 = -1/(-1/2)  = 2

c1/c2 = -2/(-1) = 2

As, a1/a2 = b1/b2 = c1/c2 = 2 .Thus, the given system of equations forms the identical lines.

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According to the question we can say that the area to the right of z = 2.00 is 0.0228 or 2.28%.

a. To find the percentage of adults in the USA with stage 2 high blood pressure, we need to find the area under the normal distribution curve to the right of 160.

Using a standard normal distribution table or a statistical software, we can find that the z-score corresponding to 160 systolic blood pressure is:

z = (160 - 117) / 22 = 2.00

The area to the right of z = 2.00 is 0.0228 or 2.28%. Therefore, around 2.28% of adults in the USA have stage 2 high blood pressure.

b. To find how many people out of a sample of 2000 would be expected to have systolic blood pressure greater than 160, we can use the same z-score from part (a) and the standard normal distribution formula:

z = (x - μ) / σ

Rearranging the formula to solve for x, we get:

x = z * σ + μ

Substituting the values, we get:

x = 2.00 * 22 + 117 = 161.4

So the expected number of people with systolic blood pressure greater than 160 out of a sample of 2000 is:

2000 * (1 - P(Z < 2.00)) = 2000 * (1 - 0.9772) = 45.6

Rounding up, we can expect about 46 people to have systolic blood pressure greater than 160 out of a sample of 2000.

c. To find the percentage of adults in the USA who qualify for stage 1 high blood pressure, we need to find the area under the normal distribution curve between 140 and 160.

Using the standard normal distribution formula and z-scores, we get:

z1 = (140 - 117) / 22 = 1.05

z2 = (160 - 117) / 22 = 1.95

Using a standard normal distribution table or a statistical software, we can find the area to the left of z1 and z2, and then subtract the two areas to get the area between z1 and z2:

P(z1 < Z < z2) = P(Z < z2) - P(Z < z1) = 0.9744 - 0.8531 = 0.1213

Therefore, approximately 12.13% of adults in the USA qualify for stage 1 high blood pressure.

d. To find the systolic blood pressure corresponding to the 30th percentile, we need to find the z-score that has an area of 0.30 to its left.

Using a standard normal distribution table or a statistical software, we can find the z-score that corresponds to the area 0.30:

z = -0.52

Using the standard normal distribution formula and the given mean and standard deviation, we can solve for the systolic blood pressure:

x = z * σ + μ = -0.52 * 22 + 117 = 105.16

Therefore, if you are in the 30th percentile for blood pressure among US adults, your systolic blood pressure is approximately 105.16.

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D. Angle Addition Postulate is necessary when proving that the angles of a parallelogram are congruent.

What is parallelogram?

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure.

To prove that opposite angles of a parallelogram are congruent, we need to use the properties of parallelograms, such as opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.

We do not need the Segment Addition Postulate, as it is used to find a length of a segment, not to prove the congruence of angles.

We also do not need to use the Opposite sides are perpendicular property, as this is only true for a rectangle or a rhombus, but not necessarily for a parallelogram.

Similarly, the Angle Addition Postulate is used to find the measure of an angle, not to prove that angles are congruent.

Therefore, the only necessary property to prove that opposite angles of a parallelogram are congruent is that opposite angles are congruent, which is a property of parallelograms.

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the correct answer is A: as x decreases to the vertical asymptote at x = -3, y decreases to negative infinity.

To determine the end behavior of the logarithmic function f(x) = log(x + 3) - 2, we need to look at what happens to the function as x approaches positive and negative infinity.

As x approaches negative infinity, the argument of the logarithm, (x + 3), becomes more and more negative. However, since logarithms are not defined for negative arguments, we need to shift the graph of the function to the left by 3 units to avoid the undefined region. This means that the vertical asymptote of the function is at x = -3. As x approaches -3 from the left, the argument of the logarithm becomes smaller and smaller negative numbers. However, the logarithm of a small negative number is a large negative number. Therefore, as x approaches -3 from the left, the function f(x) = log(x + 3) - 2 decreases to negative infinity. This eliminates options C and D.

As x approaches positive infinity, the argument of the logarithm, (x + 3), becomes more and more positive. Therefore, as x approaches positive infinity, the logarithm of (x + 3) becomes larger and larger. This means that the function f(x) = log(x + 3) - 2 approaches infinity as x approaches positive infinity. However, it approaches infinity from below since we subtract 2 from the logarithmic value.

To summarize, as x approaches -3 from the left, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches infinity from below. Therefore, the correct answer is A: as x decreases to the vertical asymptote at x = -3, y decreases to negative infinity.

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Debido a restricciones de longitud, los límites de la función aparecen descritos en la explicación de esta pregunta.

En este problema debemos determinar los límites de un punto y los límites laterales de la función. El limite de un punto se determina mediante la evaluación directa de la función, mientras los límites laterales de la función se determinan viendo a que valor tiende la función.

A continuación, determinamos los límites de las funciones:

Caso 1:

Subcaso A (x = 1)

f(x) = - 1

Subcaso B (x = 5)

f(x) = 1

Subcaso C (x = 3⁻)

f(x) → - ∞

Subcaso D (x = 3⁺)

f(x) → + ∞

Subcaso E (x = 3)

No existe

Case 2:

Subcaso A (x = 0)

f(x) = 0

Subcaso B (x = - 2)

f(x) = 0.5

Subcaso C (x = 3⁻)

f(x) → - ∞

Subcaso D (x = - 3⁺)

f(x) → + ∞

Subcaso E (x = - 3)

No existe

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Answer:

a) Area of STAR = 48 square units

b) Area of SKCO = 42 square units

Step-by-step explanation:

The formula for the area of a trapezoid is half the sum of the bases multiplied by the height:

[tex]\boxed{\sf Area=\dfrac{a+b}{2} \cdot h}[/tex]

The bases of a trapezoid are the parallel sides.

The height of a trapezoid is the perpendicular distance between the two bases.

The bases are parallel sides SR and TA.

The height is the perpendicular distance between SR and TA.

Therefore:

Substitute these values into the formula and solve for area:

[tex]\begin{aligned}\sf \implies Area\;STAR&=\dfrac{4+8}{2} \cdot 8\\\\&=\dfrac{12}{2} \cdot 8\\\\&=6\cdot 8\\\\&=48\;\sf square\;units\end{aligned}[/tex]

The bases are parallel sides SK and OC.

The height is the perpendicular distance between SK and OC.

Therefore:

Substitute these values into the formula and solve for area:

[tex]\begin{aligned}\sf \implies Area\;SKCO&=\dfrac{4+10}{2} \cdot 6\\\\&=\dfrac{14}{2} \cdot 6\\\\&=7 \cdot 6\\\\&=42\;\sf square\;units\end{aligned}[/tex]

Answer:

This is the answer of that question

Answer:

a) 4.46 miles

b) 3 miles

Step-by-step explanation:

Law of Sines:

[tex]\dfrac{\text{a}}{\text{sin(A)}} =\dfrac{\text{b}}{\text{sin(B)}}[/tex]

a) The distance of the plane from point A

The angle of depression corresponds to the congruent angle of elevation therefore, 180 - 28 - 52 = 100°

[tex]\dfrac{\text{6.7}}{\text{sin(100)}} =\dfrac{\text{b}}{\text{sin(41)}}[/tex]

[tex]\text{b}=\dfrac{6.7\text{sin}(41)}{\text{sin}(100)}[/tex]

[tex]\text{b}=4.46 \ \text{miles}[/tex]

b) Elevation of the plane

[tex]\text{sin}=\dfrac{\text{opposite}}{\text{hypotenuse}}[/tex]

hypotenuse is 4.46 and opposite is the elevation(h) to be found

[tex]\text{sin}(38)=\dfrac{\text{h}}{4.46}[/tex]

[tex]\text{h}=\text{sin}(38)4.46[/tex]

[tex]\text{h}=3[/tex]

The equation in slope-intercept form for the line that passes through the point (-8,7) and has a slope of 3/2 is y = (3/2)x + 19.

The expression [tex]x^{5}[/tex]/[tex]x^{10}[/tex] can be written in two forms: Exponent form, which is simplified and has a base of x, and Fraction form, which has a numerator and denominator separated by a horizontal line.

Exponents are a shortened notation for repeated multiplication of a number, and exponent form is a means of formulating a mathematical equation using exponents. A superscript (a little raised number) is written to the right of a number or variable that has been raised to a power in exponent form.

There are two distinct but equal ways to write the expression "[tex]x^{5}[/tex]/[tex]x^{10}[/tex]":

Exponent form: In this format, the exponents of the equation are consolidated and made simpler. We may subtract the exponents in the denominator from the exponents in the numerator since the bases of the numerator and denominator are both x:

[tex]x^5/x^10 = x^(5-10) = x^(-5)[/tex]

Fraction form: In this format, the equation is expressed as a fraction with a horizontal line between the numerator and denominator. X is increased to the fifth power in the denominator and to the tenth power in the numerator:

[tex]x^5/x^10[/tex]

X is increased to the fifth power, or x, in the numerator. X is multiplied by 10 to create the denominator, or [tex]x^{10}[/tex].

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The solution of given derivative of y = √(arctan(x)) is:

 Y' = 1 / (2√arctan(x)(1 + x²))

In general, the term "differentiate" means to distinguish or recognize the differences between two or more things or concepts.

In mathematics, differentiation refers to the process of finding the rate at which a function changes with respect to one of its variables. It is a fundamental concept in calculus and involves calculating the derivative of a function. The derivative gives us the slope of a tangent line to the curve of the function at a specific point

To differentiate Y = √arctan(x), we need to use the chain rule and the formula for differentiating the arctan function, which is:

d/dx arctan(x) = 1 / (1 + x²)

Using the chain rule, we have:

Y = √arctan(x)

Y = [tex](arctan(x))^(1/2)[/tex]

Y' =[tex](1/2)(arctan(x))^(-1/2) * d/dx (arctan(x))[/tex]

Y' = [tex](1/2)(arctan(x))^(-1/2) * (1 / (1 + x^{2} ))[/tex]

Putting it all together, we have:

Y' = (1 / 2√arctan(x)) * (1 / (1 + x²))

Simplifying the expression, we get:

Y' = 1 / (2√arctan(x)(1 + x²))

Therefore, the derivative of Y = √arctan(x) is:

Y' = 1 / (2√arctan(x)(1 + x²))

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The lateral surface area of the triangular prism is 275.5 sq. mm.

"Being to the side" is the definition of the term "lateral." A triangular prism's lateral area is equal to the sum of its side faces' areas (which are 3 rectangles). Specifically, it is the total surface area less the surface areas of the two bases. Also called the lateral surface area (LSA). Due to the two dimensions involved in its calculation, we measure it in square units.

The lateral surface area of the triangular prism is given as:

LSA = (a + b + c)h

Here, the value of a = 7.5, b = 10.5, and c = 11mm, and h = 9.5mm.

Substitute the values:

LSA = (7.5 + 10.5 + 11) (9.5)

LSA = 275.5 sq. mm

Hence, the lateral surface area of the triangular prism is 275.5 sq. mm.

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On solving the provided question we can say that As a result, the value angles of x is 23.

An angle is a shape in Euclidean geometry that is made up of two rays that meet at a point in the middle known as the angle's vertex. Two rays may combine to form an angle in the plane where they are located. When two planes collide, an angle is generated. They are referred to as dihedral angles. An angle in plane geometry is a possible configuration of two radiations or lines that express a termination. The word "angle" comes from the Latin word "angulus," which meaning "horn." The vertex is the place where the two rays, also known as the angle's sides, meet.

m(arc GH) = 122° plus 5x plus 7

5x + 129 = m(arc GH)

When we plug this into the equation we discovered earlier, we get:

5x + 7 = ½

(5x + 129)

10x + 14 = 5x + 129

5x = 115

x = 23

As a result, the value of x is 23.

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The solution of the equation is:  dy/dx = -3[tex]e^{3x}[/tex]/ √(1-[tex]e^{6x}[/tex])

What is the solution?

Let's start by using the chain rule to find dy/dx:

dy/dx = dy/du * du/dx

where u = [tex]e^{3x}[/tex]

We can find du/dx using the power rule:

du/dx = 3[tex]e^{3x}[/tex]

Now we need to find dy/du. We can use the derivative of arccos function:

dy/du = -1/√(1-u²)

Substituting u = [tex]e^{3x}[/tex], we get:

dy/dx = dy/du * du/dx

dy/dx = -1/√(1-[tex]e^{6x}[/tex]) * 3[tex]e^{3x}[/tex]

So the final answer is:

dy/dx = -3[tex]e^{3x}[/tex] / √(1-[tex]e^{6x}[/tex])

what is chain rule?

The chain rule is a fundamental rule of calculus that allows you to find the derivative of a composite function. A composite function is a function that is formed by taking one function and plugging it into another function.

For example, if we have two functions f(x) and g(x), the composite function is defined as:

h(x) = f(g(x))

To find the derivative of the composite function h(x), we use the chain rule, which states that:

h'(x) = f'(g(x)) * g'(x)

In words, this means that to find the derivative of the composite function, we first take the derivative of the outer function f(x) with respect to its input, evaluated at the inner function g(x), and then multiply it by the derivative of the inner function g(x) with respect to its input.

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Complete question is:  dy/dx = -3[tex]e^{3x}[/tex]/ √(1-[tex]e^{6x}[/tex])

Answer:

13.2 miles

Step-by-step explanation:

We can use the following conversion factors:

1 mile = 5,280 feet

Using this conversion, we can divide the height of the volcano in feet by 5,280 to get the height in miles:

69,657.652 ft ÷ 5,280 ft/mi ≈ 13.2 mi

Therefore, the height of the volcano on the recently discovered planet is approximately 13.2 miles.

Hopes this helps

Answer:

The phase of the business cycle that would be marked by an increase in productivity while employment and profits also rise is known as the Expansion phase.

Answer:

Step-by-step explanation:

A] five as 20 divided by four is five

Point K and F

Looking at the data, you can see that most of the data points form close to a line- if you were to draw a line of best fit through the data, you would see that it would be quite close to most points on the graph. This is the trend of the graph, it is linear.

However, you can see thaf K and F do not come close to this trend at all and are therefore outliers since they do not fall close to the line.

the overall score of Player A in the 2014 golf tournament was -15.

In golf, scores that are below par are represented with negative numbers. Therefore, to calculate the overall score, we need to add up all the scores.

Let's call the scores for each round A1, A2, A3, and A4, respectively. We can write the given scores as:

A1 = -4

A2 = -4

A3 = -4

A4 = -3

To find the overall score, we add up all the scores:

Overall score = A1 + A2 + A3 + A4

Overall score = (-4) + (-4) + (-4) + (-3)

Overall score = -15

Therefore, the overall score of Player A in the 2014 golf tournament was -15.

It's worth noting that in golf, the winner is the player who has the lowest score or the most negative score. So, in this case, Player A had the lowest score, making her the winner of the tournament.

In general, the overall score in golf can be calculated by adding up the scores for each hole or round. The player with the lowest score at the end of the tournament is declared the winner. Scores can be compared between different golfers or rounds, regardless of the difficulty of the course or weather conditions, by using the number of strokes above or below par.

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Answer:

r = 8 units

Step-by-step explanation:

The equation of the circle centered at the origin is x² + y² = r²

   x² + y² = 64

   x² + y² = 8²

      r = 8 units

Answer:

8

Step-by-step explanation:

An equation of a circle is depicted by the notation [tex](x-h)^2+(y-k)^2=r^2[/tex], where

- [tex](h,k)[/tex] is the center point of the circle

- [tex]r[/tex] is the radius (the span of units from the origin to the border of the circle)

In this unique problem, the center is the origin (0,0), so [tex]h[/tex] and [tex]k[/tex] respectively equal 0.

Since the equation of the circle is [tex]x^2+y^2=64[/tex] , we must square root 64 since it's in it's [tex]r^2[/tex] form to obtain the value of [tex]r[/tex], which is the radius.

[tex]\sqrt{64}[/tex]=±8

Despite mathematically, -8 or 8 will work in this case, you cannot have a negative radius; therefore, the radius of the circle is 8.

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The transmitters are approximately 115.1 miles apart.

This is a trigonometry problem that can be solved using the Law of Cosines.

We can use the law of cosines to solve this problem. Let's call the distance between the transmitters "d". Then we have:

d^2 = 115^2 + 140^2 - 2(115)(140)cos(132°)

d^2 = 13212.4

d ≈ 115.1 miles (rounded to the nearest tenth)

Therefore, the transmitters are approximately 115.1 miles apart

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Answer:

Step 1: Find the previous balance.

After 8 payments, the remaining balance on the loan is $2,237.27. Therefore, the previous balance would be the balance before the 8th payment.

Total number of payments = 24

Number of payments already made = 8

Number of payments remaining = 24 - 8 = 16

Using the given monthly payment, we can find the balance before the 8th payment:

PV = PMT x [(1 - (1 + r/n)^-n*t)/(r/n)]

where PV is the present value or previous balance, PMT is the monthly payment, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time period in years.

Plugging in the given values, we get:

PV = $147.37 x [(1 - (1 + 0.075/12)^(-12*2))/(0.075/12)]

PV = $3,090.60

Therefore, the previous balance was $3,090.60.

Step 2: Find the interest for the 9th month.

We know that the balance at the end of the 8th month was $2,237.27. We can use this and the given interest rate to find the interest for the 9th month.

Interest = Balance x (Annual interest rate/12)

Interest = $2,237.27 x (0.075/12)

Interest = $13.99

Step 3: Find the final payment.

To find the final payment, we need to add the interest for the 9th month to the monthly payment and subtract it from the remaining balance.

Final payment = Remaining balance + Interest for 9th month - Monthly payment

Final payment = $2,237.27 + $13.99 - $147.37

Final payment = $2,103.89

Therefore, the final payment is $2,103.89.

Hope This Helps!

Step-by-step explanation:

The SSS (Side-Side-Side) Congruence Postulate states that if three sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent.

To create your own SSS Congruence Postulate, you could use the following statement:

"If the corresponding sides of two triangles are congruent, then the triangles are congruent."

This postulate would encompass the SSS Postulate, as well as two other postulates: the SAS (Side-Angle-Side) Postulate and the ASA (Angle-Side-Angle) Postulate.

Using this postulate, you could show that two triangles are congruent if, for example, their corresponding sides are all 5cm long. This would mean that the triangles have the same shape and size, even if they are oriented differently.

Hopes this helps

Answer:

50 spherical candles can be made with 152,604 cm³ of wax.

Step-by-step explanation:

Since the wax candles are in the shape of a solid sphere, we can calculate the volume of one candle by using the volume of a sphere formula.

[tex]\boxed{\begin{minipage}{4 cm}\underline{Volume of a sphere}\\\\$V=\dfrac{4}{3} \pi r^3$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

The diameter of a sphere is twice its radius.

Therefore, if the diameter of the spherical candle is 18 cm, its radius is:

[tex]\implies r=\dfrac{d}{2}=\dfrac{18}{2}=9\; \sf cm[/tex]

Substitute r = 9 and π = 3.14 into the formula to calculate the volume of one spherical candle.

[tex]\begin{aligned}\implies \textsf{Volume of one candle}&=\sf \dfrac{4}{3} \cdot 3.14 \cdot (9\;cm)^3\\\\&=\sf \dfrac{4}{3} \cdot 3.14 \cdot 729\;cm^3\\\\&=\sf 3052.08\; \sf cm^3\end{aligned}[/tex]

Given the company has a total of 152,604 cm³ of wax, to calculate how many candles can be made, divide the total amount of available wax by the wax needed to make one candle.

[tex]\begin{aligned}\textsf{Total number of candles}&=\sf \dfrac{152604\; cm^3}{3052.08 \;cm^3}\\\\&=\sf 50\end{aligned}[/tex]

Therefore, 50 spherical candles can be made with 152,604 cm³ of wax.

According to the given condition, the correct expressions is:

cotαcotβ−1

Trigonometric equations are mathematical equations that involve trigonometric functions, such as sine, cosine, tangent, cotangent, secant, or cosecant, and their variables. Trigonometric equations typically involve finding the values of the unknowns that satisfy the given equation, subject to certain restrictions or conditions on the domain of the trigonometric functions.

According to the given information:

Using trigonometric identities, we can simplify the expression cos(α−β)cosαcosβ:

cos(α−β)cosαcosβ = cos(α−β) * (cosα * cosβ)

Next, we can use the identity cos(α−β) = cosαcosβ + sinαsinβ to substitute into the expression:

cos(α−β)cosαcosβ = (cosαcosβ + sinαsinβ) * (cosα * cosβ)

Now, we can distribute and simplify:

cos(α−β)cosαcosβ = cosα² * cosβ² + sinαsinβ * cosα * cosβ

Finally, using the identity cotα = 1/tanα, we can rewrite the expression as:

cos(α−β)cosαcosβ = cotαcotβ - 1

So, the equivalent expression is cotαcotβ - 1.

Therefore, according to the given condition. The correct answer is:

cotαcotβ−1

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