a card is drawn randomly from a standard 52-card deck. find the probability of the given event. write your answers as reduced fractions or whole numbers.

We cannot construct an interval using the normal approximation of survival rate between control and treatment groups because the samples must be random, independent, and their sample sizes must be sufficiently large.

The normal approximation is valid when the sample sizes are large enough to ensure that the sampling distribution of the mean of the variable is approximately normal.

The central limit theorem applies to the distribution of the sample mean when the sample size is large enough, according to the normal approximation.

As a result, the mean difference between the two groups must have a normal distribution. The normal distribution may not be an accurate representation of the underlying distribution of the difference between the two population means in the absence of this requirement, causing the confidence interval to be inaccurate. It will lead to incorrect inferences about the difference in the survival rates of the two groups.

The confidence interval constructed despite this problem will lead to incorrect inferences about the difference in the survival rates of the two groups. This would make it difficult to draw any conclusions based on the findings of this experiment.

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The measure of angle G is 34° because the sum of the measures of the angles GFE and GF is 180° - 28° = 152° and the measure of angle GFE is 28°, so the measure of angle GF is 152° - 28° = 34°.

An angle is a shape formed when two lines or rays intersect. It is measured in degrees and is used in mathematics and geometry to describe the size and direction of a corner or turn. Angles measure the amount of turn between two lines, and can be described as either acute, obtuse, right, or reflex. Angles are also used to describe the orientation of objects in space, such as the position of the sun or the direction of a car relative to its surroundings.

The measure of angle GFE is 180° minus angle FEG, which is 180° - 28° = 152°. Since the sum of the measures of the angles of a triangle must equal 180°, the measure of angle GF is 180° - 152° = 28°. Therefore, the measure of angle G is 34° because the sum of the measures of the angles GFE and GF is 180° - 28° = 152° and the measure of angle GFE is 28°, so the measure of angle GF is 152° - 28° = 34°.

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The ∠E is 28°. Then what is the angle that is on the opposite side of ∠E It is ∠G.∠G would be 28°.

An angle is a shape formed when two lines or rays intersect. It is measured in degrees and is used in mathematics and geometry to describe the size and direction of a corner or turn. Angles measure the amount of turn between two lines, and can be described as either acute, obtuse, right, or reflex. Angles are also used to describe the orientation of objects in space, such as the position of the sun or the direction of a car relative to its surroundings.

198: line EF and line FG have same length of 6.

A triangle that has to be two sides of the equal length is the  called isosceles triangle.

So your triangle is an isosceles triangle!

An isosceles with the  triangle has two angles on facing each other are the same.

For example, the ∠E is 28°. Then what is the angle that is on the opposite to side of ∠E? It is ∠G.

∠G would be 28°.

So your answer is C. 28°.

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Using coefficients ak, the following product may be extended into a power series: the expression is provided in the attached file. For each of the [tex]k 2 N[/tex]phrases, determine the coefficients ak before them. The formula [tex]ak = (-1)k(k+1)/2[/tex] yields the coefficients ak.

To get the coefficients ak, we may first simplify the above formula by factoring out a -x and rearranging terms. This results in the equation: [tex](1-x)/(1+x)2 = -x/(1+x) - x2/(1+x)2.[/tex]

Now, each term in the statement may be expanded into a power series using the formula for the geometric series. This results in: Both[tex]-x/(1+x) and -x2/(1+x)2[/tex] are equal to[tex]-x + x + x + 2 + x + 3 +...[/tex]

By combining like terms and adding these two power series, we can determine that the coefficient in front of [tex]xk is (-1)k(k+1)/2.[/tex]  Hence,[tex]ak = (-1)k(k+1)/2[/tex] is the formula for the coefficients ak.

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Step-by-step explanation:volume = 0.5 * b * h * length , where b is the length of the base of the triangle, h is the height of the triangle, and length is prism length

The population increase according to the given years will be 27500, 23750 and 20833.

Mathematicians use multiplication to calculate the product of two or more integers. It is a fundamental operation in mathematics that is frequently used in everyday living. When we need to combine sets of similar sizes, we use multiplication. The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the factors that are compounded are referred to as the factors. Repeated adding of the same number is made easier by multiplying the number.

In this question,

p = 200,000 3/4t

When t=0

p=0

When t=1

p= 20000 3/4= 27500

when t=2

p= 20000 3/8 = 23750

When t=3

p= 20000 3/12 = 20833

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The distance of the line is √17 and the midpoint of the line is ( 5/2 , 2).

Why do you use the word distance?

Distance is the total movement of an object without taking direction into account. Distance can be described as the amount of ground that an object has covered, regardless of its beginning or ending points.

points A (3,-1) B (2,5)

D = √(x₂ - x₁)² + (y₂ - y₁ )²

    = √(2 - 3)² + ( 5- ( -1))²

     = √(-1)² + (4)²

      = √1 + 16

        = √17

  midpoint =   x = 2 + 3/2

                          = 5/2

                   y = 5 - 1/2

                   y = 2

  (x,y) = ( 5/2 , 2)

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The system of inequalities to represent the constraints of the situation are x ≤ 260, y ≤ 320, and x + y ≤ 360.

A group of two or more linear inequalities are grouped together and graphed on a coordinate plane to discover the solution that concurrently solves all of the inequalities. Each inequality forms a half-plane on the coordinate plane, and the location where all the half-planes overlap is where the system is solved. Each point inside the feasible area meets all of the system's inequalities. This region is known as the feasible region. To identify the optimum solution given a set of constraints, systems of linear inequalities are frequently utilised in optimization issues.

Let us suppose the number of boards of Mahagony sold = x.

Let us suppose the number of black walnut boards sold = y.

According to the given problem the equation can be set as follows:

x ≤ 260 (the company has 260 boards of mahogany available)

y ≤ 320 (the company has 320 boards of black walnut available)

x + y ≤ 360 (the company expects to sell at most 360 boards of wood)

Hence, the system of inequalities to represent the constraints of the situation are x ≤ 260, y ≤ 320, and x + y ≤ 360.

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The value of m that makes u and v parallel vectors is 5.

if u and v are parallel, we need to find a scalar k such that:

u = kv

k(3i) = I (the i-component of u is equal to k times the i-component of v)

k(-6j) = -aj (the j-component of u is equal to k times the j-component of v, but with a negative sign since the j-component of u is negative)

k(mk) = 5k (the k-component of u is equal to k times the k-component of v)

Simplifying each equation, we get:

[tex]3k = 1\\-6k = -a\\m*k = 5k[/tex]

From the first equation, we get k = [tex]\frac{1}{3}[/tex]. Substituting this value into the second equation, we get a = 2. Finally, substituting k = [tex]\frac{1}{3}[/tex] into the third equation, we get:

m*([tex]\frac{1}{3}[/tex]) = 5*[tex]\frac{1}{3}[/tex])

Simplifying, we get:

m = 5

Parallel vectors are two or more vectors that have the same or opposite direction, regardless of their magnitudes. In other words, parallel vectors are vectors that are either collinear or antiparallel. To determine if two vectors are parallel, one can check if the cross product of the vectors is zero. If the cross product is zero, then the vectors are parallel. If the cross product is nonzero, then the vectors are not parallel.

Parallel vectors are commonly used in various fields such as physics, engineering, and computer science. For example, in physics, parallel vectors are used to describe the motion of objects along a straight line, while in computer graphics, they are used to represent the direction of light and the orientation of objects in a scene.

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In the context of engineering and construction, a bearing point refers to a specific location or area. The bearing of A from C is 230°.

The load from the structural element is transferred to the foundation at a bearing point, which is often a concentrated load point.

To avoid structural failure, settling, or excessive deflection of the part, it is crucial to make sure the bearing point is properly designed and supported.

Since the bearing of C from A is 050°, we can find the bearing of A from C by adding 180° to 50°, which gives us:

Bearing of A from C = 50° + 180° = 230°

Therefore, In the context of engineering and construction, a bearing point refers to a specific location or area. The bearing of A from C is 230°.

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Complete question -

When a 666-sided fair die is rolled, then the probability of rolling greater than 4 is  0.9940 or 99.40%.

Given that the die is fair and has 666 sides. So, each face of the die will have a probability of 1/666, i.e.,

p(1) = p(2) = ... = p(666) = 1/666.

The probability of rolling greater than 4 is P(roll greater than 4), which is the sum of the probabilities of rolling a 5, 6, 7, 8, 9, ..., 666. So,

P(roll greater than 4) = p(5) + p(6) + p(7) + ... + p(666)

P(roll greater than 4) = (1/666) + (1/666) + (1/666) + ... + (1/666)

(There are 661 terms)P(roll greater than 4) = 661(1/666)

P(roll greater than 4) = 0.9940 (rounded to four decimal places)

Hence, the probability of rolling greater than 4 is 0.9940 or 99.40%.

Alternatively, the probability of rolling greater than 4 is

   1 - P(roll less than or equal to 4)P(roll greater than 4)

= 1 - P(roll less than or equal to 4)P(roll greater than 4)

= 1 - (p(1) + p(2) + p(3) + p(4))P(roll greater than 4)

= 1 - (4/666)P(roll greater than 4)

= 1 - 0.0060P(roll greater than 4)

 = 0.9940 (rounded to four decimal places)

Hence, the probability of rolling greater than 4 is 0.9940 or 99.40%.

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A prism with a rectangular base is a figure that has two parallel and congruent rectangular bases connected by rectangular or parallelogram-shaped lateral faces.

Therefore, the figure that is a prism is "a prism with a rectangular base."

A prism with a rectangular base is a three-dimensional solid figure that has two parallel and congruent rectangular bases connected by rectangular or parallelogram-shaped lateral faces. It is a type of prism, which is a geometric figure that has identical ends and flat sides that connect them.

A pyramid with a rectangular base has a rectangular base and triangular faces that meet at a single vertex. A cylinder has two congruent circular bases and a curved lateral surface connecting them. A pyramid has a polygonal base and triangular faces that meet at a single vertex.

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Answer: 1.45 Cents per (Rounded to the nearest cent)

Step-by-step explanation:

26 apples = $37.70

We want what one apple costs individually. The best way to do this is to divide both sides by 26.

1 apple = 37.70/26

1 apple = 1.45 Cents

c ccccccccccccccccccccccccccccc

The range of the quadratic function above is (-∞, 7].

In mathematics, a quadratic prοblem is οne that invοlves multiplying a variable by itself, alsο knοwn as squaring. In this language, the area οf a square is equal tο the length οf its side multiplied by itself. The term "quadratic" cοmes frοm the Latin wοrd fοr square, quadratum.

Tο determine the quadratic functiοn's range, we must first determine the functiοn's minimum and maximum pοints. The given functiοn is in vertex fοrm, with the vertex at the pοint (h, k), where h is the vertex's x-cοοrdinate and k is the vertex's y-cοοrdinate.

We can see frοm the given equatiοn that the vertex is at the pοint (1, 7). Because the cοefficient οf the x² term is pοsitive, the parabοla οpens upwards and the vertex is the functiοn's minimum pοint.

Thus, The range of the quadratic function above is (-∞, 7].

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bobby is hanging a cabinet, the cabinet is 3.5 feet wide and 2 feet tall, if he wants to center the cabinet horizontally on a wall that is 6.25 feet wide, The end of the cabinet will be 1.375 feet away from the edge of the wall.

Bobby is hanging a cabinet. The cabinet is 3.5 feet wide and 2 feet tall. If he wants to center the cabinet horizontally on a wall that is 6.25 feet wide, how far will the end of the cabinet be from the edge of the wall? Bobby is hanging a cabinet that is 3.5 feet wide and 2 feet tall.

If he wants to center the cabinet horizontally on a wall that is 6.25 feet wide, how far will the end of the cabinet be from the edge of the wall?A cabinet is 3.5 feet wide and needs to be centered horizontally on a wall that is 6.25 feet wide. Therefore, the space remaining on the wall is: 6.25 ft - 3.5 ft = 2.75 ft.

So, the amount of space remaining on either side of the cabinet is 2.75 ft / 2 = 1.375 ft.The end of the cabinet will be 1.375 feet away from the edge of the wall.

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Answer:

$735.264

Step-by-step explanation:

First, we need to find 8% of the total payment of the TV:

8% of $799.20

[tex]\frac{8}{100}[/tex] x $799.20

= $63.936

Finally, subtract $63.936 from $799.20:

= $735.264.

Therefore, if I bought a TV and the tax rate is 8% and the total payment of the TV is $799.20, the before tax of the TV is $735.264.

Answer:

adjacent = cos(angle) x hypotenuse

Answer: 61 kg

Step-by-step explanation:

To find the average mass of the other 5 people, we need to subtract the mass of the lightest person from the total mass of all six people and then divide by 5 (since we're looking for the average of the other 5 people). Here are the steps:

   Find the total mass of all six people:

   To find the total mass of all six people, we can multiply the average mass by 6:

Total mass of all six people = 58 kg/person x 6 people = 348 kg

   Subtract the mass of the lightest person:

   We need to subtract the mass of the lightest person (43 kg) from the total mass of all six people:

Total mass of the other 5 people = Total mass of all six people - Mass of the lightest person

Total mass of the other 5 people = 348 kg - 43 kg = 305 kg

   Find the average mass of the other 5 people:

   Finally, we divide the total mass of the other 5 people by 5 to find the average mass:

Average mass of the other 5 people = Total mass of the other 5 people / 5

Average mass of the other 5 people = 305 kg / 5 = 61 kg

Therefore, the average mass of the other 5 people is 61 kg.

The value of x will be 9.

What is Transversal?

In geometry, a transversal is a line that intersects two or more other lines in a plane. When a transversal intersects two parallel lines, it creates eight angles, four on each side of the transversal. The angles that are opposite to each other and not next to each other are called alternate angles, while the angles that are on the same side of the transversal and not next to each other are called corresponding angles. The angles that are next to each other and on the same side of the transversal are called adjacent angles, and the angles that are opposite to each other and next to each other are called vertical angles.

We know that if two transversal cuts the parallel lines, then the ratio of length of corresponding sides is equal.

So, we have,

    8 / (x-3) = 4 / 3

Now, we can solve for x as follows:

  8 / (x-3) = 4 / 3

  8 × 3 = 4 (x-3)

  24 = 4x - 12

  24 + 12 = 4x

  36 = 4x

 ∴ x = 9.

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The monthly payment on a 6-year loan of $15,250 at 3.5% compounded monthly is (b)$235.13 (rounded to the nearest cent).

To find the monthly payment on a loan, we can use the formula:

PMT = (P × r) / [1 - (1 + r) ^ -n]

Where: P = principal amount (in this case, $15,250)

r = interest rate per period (monthly rate = 3.5% / 12 ⇒ 0.002917)

n = the total number of periods (6 years × 12 months/year ⇒ 72 months)

Now we can substitute the values:

PMT = ($15,250 × 0.002917) / [1 - (1 + 0.002917) ^ -72]

After solving we get:

PMT ≈ $235.125 → rounded to the nearest cent,

The compound monthly payment is $235.13.Therefore, option (b) is correct.

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The rates for the following term in the year a, is [tex]365[/tex], part b, is [tex]189[/tex], part c's difference is unclear, and part d's rate for the final term is [tex]123[/tex].

A term is the name given to each integer in a series. A series has a place for each phrase. Think about the order, for instance Each number in the series is referred to as a word.

The nth term formula, where stood for the term number, can be used to locate any term in a series. Formulas: An arithmetic sequence's nth term is represented by the formula: a n = a + n - 1 d, where is the first word and is a clear differentiation.

(a) To find the rate for the next term in the sequence [tex]2, 5, 14, 41, 122[/tex]

[tex]122 + 3(81) = 365[/tex]

The next term in the sequence is [tex]365[/tex].

(b) To find the rate for the next term in the sequence [tex]1, 5, 13, 29, 61[/tex]

[tex]61 + 4(32) = 189[/tex]

So the next term in the sequence is [tex]189[/tex].

(c) To find the rate for the next term in the sequence [tex]1, 212, 34, 78, 166[/tex]  the differences between consecutive terms are not following a clear pattern. Therefore, we cannot determine the rate for the next term with the information given.

(d) To find the rate for the next term in the sequence[tex]6, 9, 15, 27, 51[/tex]

[tex]51 + 3(24) = 123[/tex]

So the next term in the sequence is [tex]123[/tex].

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For every consecutive solid that is dilated by k, the volume is multiplied by k³ thus, Elena's reasons can be used to argue the reasoning.

A dilatation is a transformation that alters the size but not the shape of an item. Every point on an item moves away from or towards a fixed position known as the centre of dilatation when the object is dilated. The scale factor is multiplied by the distance between each location and the centre of dilatation.

Thinking about a cube with s sides. We are aware that this cube's volume is s³. This cube will have a new side length of ks and a new volume of (ks)³ = k³s³ if we dilate it by a factor of k. Hence, as Clare said, the volume has been doubled by k³.

Fo any other solid right now. Similar to how a 2D form may be thought of as being composed of several little rectangles, we can conceive of this solid as being composed of many small cubes. Each of these little cubes will be dilated by a factor of k if we dilate the solid by a factor of k.

Hence, for any solid, if it's dilated by a factor of k, the volume is multiplied by k³.

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The Trapezoidal rule and Simpson's rule are two methods used to approximate the value of a definite integral. The Trapezoidal rule approximates the integral by dividing the region between the lower and upper limits of the integral into n trapezoids, each with a width h. The approximate value of the integral is then calculated as the sum of the areas of the trapezoids. The Simpson's rule is similar, except the region is divided into n/2 trapezoids and then the integral is approximated using the weighted sum of the area of the trapezoids.

For the given integral 4 x x2 1 0 dx, with n = 200, the Trapezoidal rule and Simpson's rule approximate the integral to be 7.4528 and 7.4485 respectively, rounded to four decimal places. The exact value of the integral is 7.4527. The difference between the exact and approximate values is very small, thus indicating that both the Trapezoidal rule and Simpson's rule are accurate approximations.

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The  value of |sin x - x| ≤ 0.5*10^-1 and the value of |sin x - x| ≤ (0.1)^2/2! = 0.005 and value of x for which these approximations hold is |x| < 0.1.

(a) Using the alternating series error bound, we can bound the error of the approximation sin x = x when |x| < 0.1 as follows:

|sin x - x| ≤ |Rn| ≤ a[n+1], where a = 1 and n = 1 since we only need to use the first term of the alternating series expansion of sin x.

Therefore, |sin x - x| ≤ 0.5*10^-1.

(b) Using the Lagrange error formula, we can bound the error of the approximation sin x = x when |x| < 0.1 as follows:

|sin x - x| ≤ |Rn| ≤ M*(x^(n+1))/(n+1)!, where M is the maximum value of the (n+1)th derivative of sin x on the interval [0, x].

In this case, we can take n = 1, and since the second derivative of sin x is bounded by 1, we have M = 1.

Therefore, |sin x - x| ≤ (0.1)^2/2! = 0.005.

(c) The value of x for which these approximations hold is |x| < 0.1.

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The solution for X in equation 30=5(X+5)X is X= 1.

To solve the equation, we can start by distributing the 5 on the right-hand side of the equation, which gives us:

30 = 5X + 25X

Combining like terms, we get:

30 = 30X

Dividing both sides by 30, we get:

X = 1

However, we need to check whether this value satisfies the original equation. Plugging X=1 into the equation gives us:

30 = 5(1+5)(1)

30 = 5(6)

30 = 30

Therefore, the only valid solution is X=1.

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The length of side x can be calculated using the Pythagorean theorem, which states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the longest side.

Length is the linear distance between two points. It is a fundamental concept in geometry, physics, and many other sciences. In mathematics, length is defined as the magnitude of a line segment, which is the distance between two points. In physics, length is the distance an object moves in a given direction, or the distance an object has traveled during a given period of time.

In this case, the longest side would be x and the two shorter sides would be 2. Therefore, x2 = 22 + 42, which simplifies to x2 = 20. This means that x = √20, which can be written in simplest radical form with a rational denominator as x = 10√2.

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According to the question the simplest radical form with a rational denominator as x = 10√2.

Length is the linear distance between two points. It is a fundamental concept in geometry, physics, and many other sciences. In mathematics, length is defined as the magnitude of a line segment, which is the distance between two points.

The length of side x can be calculated using the Pythagorean theorem, which states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the longest side.
In this case, the longest side would be x and the two shorter sides would be 2.
Therefore, x2 = 22 + 42, which simplifies to x2 = 20. T
his means that x = [tex]\sqrt{20}[/tex], which can be written in simplest radical form with a rational denominator as [tex]x = 10 \sqrt{2}[/tex]

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tan nx + tanx1 - tan nxtanx =tan(n+1)x identity holds true in two cases.

To prove the identity:

tan(nx) + tan(x1) - tan(nx)tan(x) = tan((n+1)x)

We'll start with the left-hand side:

tan(nx) + tan(x1) - tan(nx)tan(x)

We can use the identity for the sum of tangents:

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

If we let A = nx and B = x1, then we can write:

tan(nx + x1) = (tan(nx) + tan(x1)) / (1 - tan(nx)tan(x1))

Simplifying the denominator:

tan(nx + x1) = (tan(nx) + tan(x1)) / (tan(nx + x1) - tan(nx)x1)

Multiplying both sides by (tan(nx + x1) - tan(nx)x1):

tan(nx + x1)(tan(nx + x1) - tan(nx)x1) = tan(nx) + tan(x1)

Expanding the left-hand side:

tan²(nx + x1) - tan(nx)x1 tan(nx + x1) = tan(nx) + tan(x1)

Moving all terms to one side:

tan²(nx + x1) - tan(nx + x1)tan(nx)x1 - tan(nx) - tan(x1) = 0

Factoring the quadratic:

(tan(nx + x1) - tan(nx)x1) (tan(nx + x1) - tan(x1)) = 0

So either:

tan(nx + x1) = tan(nx)x1

Or:

tan(nx + x1) = tan(x1)

If we consider the case where tan(nx + x1) = tan(nx)x1, then we can substitute this expression into the left-hand side of the original identity:

tan(nx) + tan(x1) - tan(nx)tan(x)

= tan(nx) + tan(x1) - (tan(nx + x1) / x1)

= tan(nx) + tan(x1) - (tan(nx)x1 / x1)

= tan(nx) + tan(x1) - tan(nx)

= tan(x1)

And this is equal to the right-hand side of the original identity, which is:

tan((n+1)x)

So the identity holds.

If we consider the case where tan(nx + x1) = tan(x1), then we can similarly substitute this expression into the left-hand side of the original identity:

tan(nx) + tan(x1) - tan(nx)tan(x)

= tan(nx) + tan(x1) - (tan(nx + x1) / x1)

= tan(nx) + tan(x1) - (tan(x1) / x1)

= (tan(nx) x1 + tan(x1) - tan(nx)tan(x)) / x1

= tan((n+1)x)

And this is equal to the right-hand side of the original identity, which is:

tan((n+1)x)

So the identity holds in this case as well.

Therefore, we have shown that the identity holds in both cases, and hence it holds in general.

What is Trigonometric identities?

Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.

There are various distinct trigonometric identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle.

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The missing area or side length in the triangles are:

1: Area = 145 units²

2: Area =  17 units²

3: Area  = 29 units²

4: Area= 27 units²

5: length = √37 units

6: length = 2√26 units

7: length = 3√11 units

8: length  = 5√3 units

Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. That is:

c² = a² + b²

Where  a and b are the lengths of the legs, and c is the length of the hypotenuse

No. 1

Area (hypotenuse)  = 81 + 64 = 145 units²

No. 2

Area (hypotenuse)  = 16 + 1 = 17 units²

No. 3

Area (hypotenuse)  = 5² + 2² = 29 units²

No. 4

Area (leg)  = 36 - 9 = 27 units²

No. 5

length (hypotenuse)  = √(6² + 1²) = √37 units

No. 6

length (hypotenuse)  = √(10² + 2²) = 2√26 units

No. 7

length (leg)  = √(10² - 1²) = 3√11 units

No. 8

length (leg)  = √(10² - 5²) = 5√3 units

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The reason why the terms a and m have to be relatively prime integers is that it is the only way to make sure that ax≡1 (mod m) is solvable for x within the integers modulo m.

Theorem:"If a and m are relatively prime integers and m > 1, then an inverse of a modulo m exists. Furthermore, this inverse is unique modulo m. (That is, there is a unique positive integer a less than m that is an inverse of a modulo m and every other inverse of a modulo m is congruent to a modulo m.)"If a and m are relatively prime integers and m > 1, then an inverse of a modulo m exists. Furthermore, this inverse is unique modulo m. (That is, there is a unique positive integer a less than m that is an inverse of a modulo m and every other inverse of a modulo m is congruent to a modulo m.)The inverse of a modulo m is another integer, x, such that ax≡1 (mod m).

This theorem has an interesting explanation: if a and m are not co-prime, then there is no guarantee that ax≡1 (mod m) has a solution in Zm. The reason for this is that if a and m have a common factor, then m “absorbs” some of the factors of a. When this happens, we lose information about the congruence class of a, and so it becomes harder (if not impossible) to undo the multiplication by .This is the reason why the terms a and m have to be relatively prime integers.

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