Establish the identity. \[ (1+\sec \theta)(1-\sec \theta)=-\tan ^{2} \theta \] Multiply and write the left side expression as the difference of two squares.

Answers

Answer 1

We have established the identity:

(1 + sec θ)(1 - sec θ) = (1 + sec θ)(tan^2θ + sec θ) = -tan^2θ

To establish the identity, let's start with the left side of the equation:

(1 + sec θ)(1 - sec θ)

We can use the identity: sec^2θ = 1 + tan^2θ

Substituting this into the expression, we have:

(1 + sec θ)(1 - sec θ) = (1 + sec θ)(1 - sec θ) = (1 + sec θ)(1 + tan^2θ)

Now, let's write the right side expression as the difference of two squares:

(1 + sec θ)(1 + tan^2θ) = (1 + sec θ)(tan^2θ + 1)

Using the distributive property, we can expand this expression:

(1 + sec θ)(tan^2θ + 1) = tan^2θ + sec θ + tan^2θ(sec θ) + sec θ

Now, simplify the expression:

tan^2θ + sec θ + tan^2θ(sec θ) + sec θ = tan^2θ(1 + sec θ) + sec θ(1 + sec θ)

Finally, notice that (1 + sec θ) is common to both terms, so we can factor it out:

tan^2θ(1 + sec θ) + sec θ(1 + sec θ) = (1 + sec θ)(tan^2θ + sec θ)

Therefore, we have established the identity:

(1 + sec θ)(1 - sec θ) = (1 + sec θ)(tan^2θ + sec θ) = -tan^2θ

Learn more about  established from

https://brainly.com/question/441178

#SPJ11


Related Questions

A contour map is shown for a function f(x,y) on the rectangle R=[−3,6]×[−1,4]. a. Use the midpoint rule with m=2 and n=3 to estimate the value of ∬R​f(x,y)dA. b. Estimate the average value of the function f(x,y). fave​≈ Hint

Answers

a. The estimated value of ∬R​f(x,y)dA is 105

b. The estimated average value of the function f(x, y) is 7.

a. The rectangle R=[−3,6]×[−1,4] is divided into m = 2 subintervals along the x-axis and n = 3 subintervals along the y-axis. Therefore, each subinterval has a width of Δx = (6 - (-3))/2 = 9/2 and a height of Δy = (4 - (-1))/3 = 5/3.

We can calculate the midpoint of each subrectangle using the formula:

[tex]x_i = x_min + (i - 0.5) * \Delta x\\y_j = y_min + (j - 0.5) * \Delta y[/tex]

where i = 1, 2, ..., m and j = 1, 2, ..., n.

Using the midpoint rule, the estimate of the double integral is given by:

∬R​f(x,y)dA ≈ Δx * Δy * ∑∑[tex]f(x_i, y_j)[/tex]

where the double summation is taken over all the midpoints (x_i, y_j) of the subrectangles.

Calculate the midpoints of the subrectangles.

[tex]x_1 = -3 + (1 - 0.5) * (9/2) = -3 + 4.5 = 1.5\\x_2 = -3 + (2 - 0.5) * (9/2) = -3 + 9 = 6\\y_1 = -1 + (1 - 0.5) * (5/3) = -1 + (1/2) * (5/3) = -1 + 5/6 = -1/6\\y_2 = -1 + (2 - 0.5) * (5/3) = -1 + (3/2) * (5/3) = -1 + 5/2 = 9/2\\y_3 = -1 + (3 - 0.5) * (5/3) = -1 + (5/2) * (5/3) = -1 + 25/6 = 19/6[/tex]

Evaluate the function at each midpoint.

[tex]f(x_1, y_1) = 2\\f(x_1, y_2) = -1\\f(x_1, y_3) = 0\\f(x_2, y_1) = 1\\f(x_2, y_2) = 3\\f(x_2, y_3) = 2[/tex]

∬R​f(x,y)dA ≈ Δx * Δy * ∑∑[tex]f(x_i, y_j)[/tex]

           = (9/2) * (5/3) * (2 + (-1) + 0 + 1 + 3 + 2)

           = (9/2) * (5/3) * 7

           = 15 * 7

           =  105

b. To estimate the average value of the function f(x, y), we can divide the double integral by the area of the rectangle R, which is A = Δx * Δy * m * n.

The average value is then given by:

f_ave ≈ (∬R​f(x,y)dA) / A

Now let's perform the calculations:

Step 1: Calculate the area of the rectangle.

A = Δx * Δy * m * n

 = (9/2) * (5/3) * 2 * 3

 = 15

Step 2: Calculate the average value.

f_ave ≈ (∬R​f(x,y)dA) / A

     = 105 / 15

     = 7

Therefore, the estimated value of ∬R​f(x,y)dA is 105 and the estimated average value of the function f(x, y) is 7.

To know more about estimated value, refer here:

https://brainly.com/question/32263011

#SPJ4

How does sample size affect determinations of statistical significance? The _________ the sample, the ________.
a. larger; greater probability that the variable has an effect
b. smaller; greater probability that the variable has an effect
c. larger; the more confident you can be in your decision to reject or retain the null hypothesis
d. smaller; the more confident you can be in your decision to reject or retain the null hypothesis

Answers

The larger; the sample, the greater probability that the variable has an effect. The correct option is (a).

When it comes to determining statistical significance, a larger sample size increases the statistical power of the analysis.

This means that with a larger sample size, there is a greater probability of detecting a true effect or relationship between variables.

This is because a larger sample size provides more information and reduces the impact of random variability.

Option (a) correctly identifies that a larger sample size leads to a greater probability that the variable has an effect. With a larger sample size, the analysis has more statistical power to detect and accurately estimate the effects or relationships being investigated.

A larger sample size also increases the precision of the estimates and reduces the sampling error, making the results more reliable and representative of the population. It allows for more accurate inference and increases the confidence in the findings.

Therefore, option (c) is also partially correct, as a larger sample size provides more confidence in the decision to reject or retain the null hypothesis.

In summary, a larger sample size improves the ability to detect effects and increases the confidence in the statistical analysis and decision-making process. The correct option is (a).

To know more about "Sampling error" refer here:

https://brainly.com/question/30765764#

#SPJ11

Jacobi wants to install an underground sprinkler system in her backyard the backyard is rectangular with side length 17 m and 26 m .the water pipe will run diagonally across the yard about how many metres of water pipe does Jacobi need .

Answers

The length of the pipe required would be 31.06 meters

The length of the pipe is the hypotenus of the triangle formed :

hypotenus = √opposite² + adjacent²

substituting the values into our equation:

length of pipe = √17² + 26²

length of pipe = √965 = 31.06

Therefore, the length of the pipe needed is 31.06 meters

Learn more on length : https://brainly.com/question/2217700

#SPJ1

4500-p² 4 The demand equation for a product is found to be a = price of the product in dollars and q is the quantity. a. Find the price elasticity of demand when the price is $40. b. Is the demand el

Answers

The demand equation for a product is a function that represents the relationship between the price of a product and the quantity demanded by consumers. The price elasticity of demand when the price is $40 is E= (40/Q) (dQ/dP) = (40/Q) (q/40) = 1 Therefore, demand is unit elastic.

The price elasticity of demand measures the responsiveness of the quantity demanded of a product to a change in its price.

It is a crucial concept in economics, particularly in understanding how consumers react to changes in prices.

To answer this question, we use the formula for price elasticity of demand:

E= (P/Q) (dQ/dP) where E is the elasticity,

P is the price of the product, Q is the quantity demanded, and

dQ/dP is the derivative of the quantity demanded with respect to the price.

Given the demand equation,

a = price of the product in dollars and q is the quantity.

Therefore, we can rewrite the equation as follows:

a = Pq Taking the derivative of both sides, we get:

da/dP

= q + P (dq/dP)  Solving for dq/dP,

we get: dq/dP

= (da/dP - q)/P

Plugging in the values, we get:

dq/dP

= (1q - 0)/40

= q/40

Hence, the price elasticity of demand when the price is $40 is

E= (40/Q) (dQ/dP)

= (40/Q) (q/40)

= 1

Therefore, demand is unit elastic.

The demand is unit elastic if the percentage change in quantity demanded is equal to the percentage change in price.

Therefore, a change in price will lead to an equal change in quantity demanded.

If the elasticity is greater than 1, the demand is elastic.

If the elasticity is less than 1, the demand is inelastic.

To know more about Equation  visit :

https://brainly.com/question/29538993

#SPJ11

The graph shows the function f(x) = |x – h| + k. What is the value of k?

Answers

The calculated value of k is -2.5

How to determine the value of k?

From the question, we have the following parameters that can be used in our computation:

The graph

(see attachment)

Also, we have

f(x) = |x - h| + k

From the graph, we have the vertex to be

(h, k) = (1, -2.5)

By comparison, we have

k = -2.5

Read mroe about functions at

https://brainly.com/question/27915724

#SPJ1

Let Y₁,..., Yn N(μ,0²). State the sampling distribution of Y = n=¹_₁ Y₁. -1 i=1 n1, Σ (Υ; – Υ)2. State the sampling distribution of S² = State the mean and variance of Y and S².

Answers

1. The sampling distribution of Y is a normal distribution with mean nμ and variance nσ².

2. The mean of the sampling distribution of S² is σ², and the variance is 2σ⁴ / (n-1).

In the given notation, Y₁, Y₂, ..., Yₙ are independent and identically distributed (i.i.d.) random variables following a normal distribution with mean μ and variance σ².

1. Sampling Distribution of Y = ∑(i=1 to n) Yᵢ:

The random variable Y represents the sum of n independent normal random variables. The sampling distribution of Y is also a normal distribution. The mean of the sampling distribution of Y can be obtained by the linearity of expectation:

E(Y) = E(∑(i=1 to n) Yᵢ) = ∑(i=1 to n) E(Yᵢ) = ∑(i=1 to n) μ = nμ

The variance of the sampling distribution of Y can be obtained by the linearity of variance:

Var(Y) = Var(∑(i=1 to n) Yᵢ) = ∑(i=1 to n) Var(Yᵢ) = ∑(i=1 to n) σ² = nσ²

Therefore, the sampling distribution of Y is a normal distribution with mean nμ and variance nσ².

2. Sampling Distribution of S²:

The random variable S² represents the sample variance calculated from a sample of n observations. The sampling distribution of S² follows a chi-square distribution with (n-1) degrees of freedom.The mean of the sampling distribution of S² is given by:

E(S²) = σ²

The variance of the sampling distribution of S² is given by:

Var(S²) = 2σ⁴ / (n-1)

Therefore, the mean of the sampling distribution of S² is σ², and the variance is 2σ⁴ / (n-1).

To know more about Mean refer here:

https://brainly.com/question/15323584#

#SPJ11

Other Questions
Q. 2 If the surface tensions of water and benzene at 20 C are 72, 28.8 dyne/ cm respectively. Find the interfacial tension? If the surface tensions of HO and C8H15OH at 20 C are 72, 17.0 dyne/ cm respectively while the interfacial tension was 10.7 dyne / cm. Calculate (i) cohesion work of C8H15OH (ii) adhesion work between HO and C8H15OH (iii) Predict if the C8H15OH will spread on the water surface or No Jacobi wants to install an underground sprinkler system in her backyard the backyard is rectangular with side length 17 m and 26 m .the water pipe will run diagonally across the yard about how many metres of water pipe does Jacobi need . Let Y,..., Yn N(,0). State the sampling distribution of Y = n=_ Y. -1 i=1 n1, (; )2. State the sampling distribution of S = State the mean and variance of Y and S. Make at least 3 suggestions according to the SWOT-PESTLE analysisin IT department How does sample size affect determinations of statistical significance? The _________ the sample, the ________.a.larger; greater probability that the variable has an effectb.smaller; greater probability that the variable has an effectc.larger; the more confident you can be in your decision to reject or retain the null hypothesisd.smaller; the more confident you can be in your decision to reject or retain the null hypothesis In the Project Opportunity Assessment, the first question is the aim of all questions.True/FalseA problem statement is an unstructured set of statements that describes the purpose of an effort in terms of what problem its trying to solve.True/False Show that the communalities in a factor analysis model are unaffected by the transformation A = AM Ex. 5.3 Give a formula for the proportion of variance explained by the jth factor estimated by the principal factor approach. He _____________ snacks most of time he is travelling. a. ate b. eaten c. eating d. eats 4500-p 4 The demand equation for a product is found to be a = price of the product in dollars and q is the quantity. a. Find the price elasticity of demand when the price is $40. b. Is the demand el Write a short two-page paper on ""blood diamonds"" and/or ""ethical diamonds."" Define each and explain the positives and negatives for this social sustainability issue. What should be the role of diamond producers? What is the role of operations managers in this industry? howto solve8. Consider the following elementary reactions (process) I) CO (g) + Cl2 (g) COCI2 (g) II) HCII (g) HCI (g) + 1 (g) What is the molecularity of each reaction and write the rate law expression Snow fields and glaciers have high ____________ and reflect 80to 90 percent of sunlight. Inverses of Functions 7. Find fg and gf, if they exist. f = {(-4,-5), (0, 3), (1,6)} and g = {(6, 1), (-5,0), (3,-4)}. 8. Find [gh] (x) and [hg](x), if they exist. g(x) = x + 6 and h(x) = 3x. 9. Find the inverse of this relation. {(-5,-4), (1, 2), (3, 4), (7,8)} 10. Find the inverse of each function. Then graph the function and its inverse. g(x) = 3 + x Question A2 Square planar metal complexes typically undergo ligand substitution via an associative mechanism, due to their low coordination number. Below is a series of ligands listed in terms of the Part 2: Short answer questions. There are 5 questions each worth 2 marks. The total mark for Part 2 is 10 marks. n databases, derived attributes are often not represented. Give two reasons why you would include derived attributes in a database? Enter your answer here Of course Sir. We .......................................... the denim you are looking for. lim (x,y)(0,0)x 2+y 29xy= A. 1 B. 1 C. 0 D. E. does not exist mevcut deil Question Find dx2d2y if x2+3y2=8 Que le hubiese dicho dorel al nio del cabello rojizo si lo hubiera encontrado en la calle? Escribir un pequeo relato en el que ellos se encuentren (capitulo IV). Es del cuento del espejo africano please help me with these twoquestions. thank you!If a tree dies and the trunk remains undisturbed for \( 1.545 \times 10^{4} \) years, what percentage of the original \( { }^{14} \mathrm{C} \) is still present? (The half-life of \( { }^{14} \mathrm{