1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Rudik [331]
3 years ago
9

Prove the following

Mathematics
1 answer:
DerKrebs [107]3 years ago
7 0

\bf [cot(\theta )+csc(\theta )]^2=\cfrac{1+cos(\theta )}{1-cos(\theta )} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{doing the left-hand side}}{[cot(\theta )+csc(\theta )]^2}\implies cot^2(\theta )+2cot(\theta )csc(\theta )+csc^2(\theta ) \\\\\\ \cfrac{cos^2(\theta )}{sin^2(\theta )}+2\cdot \cfrac{cos(\theta )}{sin(\theta )}\cdot \cfrac{1}{sin(\theta )}+\cfrac{1}{sin^2(\theta )}\implies \cfrac{cos^2(\theta )}{sin^2(\theta )}+\cfrac{2cos(\theta )}{sin^2(\theta )}+\cfrac{1}{sin^2(\theta )}

\bf \cfrac{\stackrel{\textit{perfect square trinomial}}{cos^2(\theta )+2cos(\theta )+1}}{sin^2(\theta )}\implies \boxed{\cfrac{[cos(\theta )+1]^2}{sin^2(\theta )}} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{doing the right-hand-side}}{\cfrac{1+cos(\theta )}{1-cos(\theta )}}\implies \stackrel{\textit{multiplying by the denominator's conjugate}}{\cfrac{1+cos(\theta )}{1-cos(\theta )}\cdot \cfrac{1+cos(\theta )}{1+cos(\theta )}}

\bf \cfrac{[1+cos(\theta )]^2}{\underset{\textit{difference of squares}}{[1-cos(\theta )][1+cos(\theta )]}}\implies \cfrac{[cos(\theta )+1]^2}{1^2-cos^2(\theta )} \\\\\\ \cfrac{[cos(\theta )+1]^2}{1-cos^2(\theta )}\implies \boxed{\cfrac{[cos(\theta )+1]^2}{sin^2(\theta )}}

recall that sin²(θ) + cos²(θ) = 1, thus sin²(θ) = 1 - cos²(θ).

You might be interested in
I need help finding the values of the last boxes shown in the image.
Bingel [31]

The volume of the region R bounded by the x-axis is: \mathbf{\iint_R(x^2+y^2)dA = \int ^{tan^{-1}(4)}_{0} \int^{\frac{2}{cos \theta}}_{0} \ r^3 dr d\theta}

<h3>What is the volume of the solid revolution on the X-axis?</h3>

The volume of a solid is the degree of space occupied by a solid object. If the axis of revolution is the planar region's border and the cross-sections are parallel to the line of revolution, we may use the polar coordinate approach to calculate the volume of the solid.

In the graph, the given straight line passes through two points (0,0) and (2,8).

Therefore, the equation of the straight line becomes:

\mathbf{y-y_1 = \dfrac{y_2-y_1}{x_2-x_1}(x-x_1)}

where:

  • (x₁, y₁) and (x₂, y₂) are two points on the straight line

Thus, from the graph let assign (x₁, y₁) = (0, 0) and (x₂, y₂) = (2, 8), we have:

\mathbf{y-0 = \dfrac{8-0}{2-0}(x-0)}

y = 4x

Now, our region bounded by the three lines are:

  • y = 0
  • x = 2
  • y = 4x

Similarly, the change in polar coordinates is:

  • x = rcosθ,
  • y = rsinθ

where;

  • x² + y² = r²  and dA = rdrdθ

Now

  • rsinθ = 0   i.e.  r = 0 or θ = 0
  • rcosθ = 2 i.e.   r  = 2/cosθ
  • rsinθ = 4(rcosθ)  ⇒ tan θ = 4;  θ = tan⁻¹ (4)

  • ⇒ r = 0   to   r = 2/cosθ
  •    θ = 0  to    θ = tan⁻¹ (4)

Then:

\mathbf{\iint_R(x^2+y^2)dA = \int ^{tan^{-1}(4)}_{0} \int^{\frac{2}{cos \theta}}_{0} \ r^2 (rdr d\theta )}

\mathbf{\iint_R(x^2+y^2)dA = \int ^{tan^{-1}(4)}_{0} \int^{\frac{2}{cos \theta}}_{0} \ r^3 dr d\theta}

Learn more about the determining the volume of solids bounded by region R here:

brainly.com/question/14393123

#SPJ1

5 0
2 years ago
Alguien puede ayudarme
AleksandrR [38]

Answer:

8

Step-by-step explanation:

simplemente encuentre el total de los 8 pagos. Hay 8 pagos ya que habrá 8 retiros totales: (2 años) × (cuatro retiros por año) = 8 retiros.

3 0
3 years ago
Joey and Armando live on the same st as the park. The park is 9/10 mile from Joey's home. Joey leaves home and walks to Armando'
AleksandrR [38]
Let's start by assuming Armando's house is between Joey's and the park. 

Let x be the distance Joey walked to Armando's house.

<span>The park is 9/10 mile from Joey's home. Joey leaves home and walks to Armando's home. Then Joey and Armando walk 3/5 mile to the park. 

</span>\dfrac{9}{10} = x + \dfrac{3}{5}

x = \dfrac{9}{10} - \dfrac{3}{5} = \dfrac{9}{10} -\dfrac{6}{10} = \dfrac{3}{10}

That's probably the answer they're looking for.  But what if the park is between Joey and Armando's houses or Joey is between the park and Armando?  (The latter isn't really possible with the given distances.)

Let a, b, c be the distances between three collinear points like we have here.  Our equation is really a few equations in one, something like

\pm a \pm b = \pm c

Let's get rid of the plus/minuses. Squaring,

a^2 + b^2\pm 2ab = c^2

\pm 2ab = c^2-a^2-b^2

4a^2b^2 = (c^2-a^2-b^2)^2

For us, that's a quadratic equation for c^2

4(9/10)^2(3/5)^2= (c^2-(9/10)^2 - (3/5)^2)^2

I'll skip right to the solutions,

c^2=\dfrac{9}{100} \textrm{ or } c^2=\dfrac{9}{4}


c=\dfrac{3}{10} \textrm{ or } c=\dfrac{3}{2}

We could have gotten the 3/2 just by adding 9/10+3/5 but this was more fun.

6 0
3 years ago
Hiii please help! i’ll give brainliest if you give a correct answer tysm
vitfil [10]

Answer:2 1/2 divided by 5/8 = ?

Step-by-step explanation:

7 0
2 years ago
There are 8 swimmers in the swimming club. The club bought c swimming caps to split among its members. Write an expression that
love history [14]

Answer:

c/8

Step-by-step explanation:

This is division. This can be figured out by reading "split". If something is to be split, it is the dividend.

5 0
2 years ago
Read 2 more answers
Other questions:
  • Given: ∠V ≅ ∠Y, bisects ∠VWY. Prove: ∆VWZ ≅ ∆YWZ Look at the figure. Which postulate or theorem proves the triangles are congrue
    13·1 answer
  • A company is in the business of finding addresses of long-lost friends. The company claims to have a 70% success rate. Suppose t
    15·1 answer
  • Marcus worked and earned $60.27 in 8.2 hours. How much did he earn per hour
    11·1 answer
  • A) Find the average rate of change of the area of a circle with respect to its radius r as r changes from 4 to each of the follo
    12·2 answers
  • What is the solution set of the equation |2x+3|=7 in the set of rational numbers?
    10·2 answers
  • Help will give crown only REAL ANSWERS COMMENT IF YOU HAVE A QUESTION
    12·1 answer
  • Find 3 consecutive even integers such that the sum of twice the smallest number and 3 times the largest is 42
    10·1 answer
  • What is the simplest form of 6/27?
    8·1 answer
  • (PLEASE HELP ASAP PLEASE EXPLAIN ALSO PLEASE AND THANK YOU)
    9·1 answer
  • Instructions: Interpret the function given in the context of the real-world situation described to answer the question.
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!