I wanted to confirm that I solved these problems correctly (we had to convert the polar curves to Cartesian equations).

1.rcos(theta)=1
x=1
2.r=2*sin(theta)+2*cos(theta)
r^2=2rsin(theta)+2rcos(theta)
x^2+y^2=2y+2x (a little unsure what do next if this is correct)
3.r=tan(theta)sec(theta)
r=sin(theta)/(cos(theta))^2
r(cos(theta))^2=sin(theta)
r^2(cos(theta))^2=rsin(theta)
x^2=y

I agree with your answers

Nice work on #3

in #2, form
x^2 + y^2 = 2y + 2x you have a circle
you could complete the square, and find its centre and radius in this way ...

x^2 + y^2 - 2y - 2x = 0
x^2 - 2x + 1 + y^2 - 2y + 1 = 2
(x-1)^2 + (y-1)^2 = 2

so centre is (1,1) and radius is √2

Let's go through each problem step by step to confirm if you solved them correctly:

1. The polar equation rcos(theta) = 1 can be converted to a Cartesian equation by using the identity x = rcos(theta). Therefore, substituting rcos(theta) with x, we get x = 1. Your answer is correct.

2. The polar equation r = 2sin(theta) + 2cos(theta) can be converted to a Cartesian equation by using the identities x = rcos(theta) and y = rsin(theta). By substituting those, we get x^2 + y^2 = 2y + 2x. Your answer is correct.

3. The polar equation r = tan(theta)sec(theta) can be simplified by using the identities r = sin(theta)/(cos(theta))^2, x = rcos(theta), and y = rsin(theta). By substituting these, we get x^2 = y. Your answer is correct.

Overall, you have solved all the problems correctly. Well done!

To confirm whether you solved these problems correctly, let's go through each one and compare your answers to the correct conversions.

1. The polar equation rcos(theta) = 1 is converted to the Cartesian equation x = 1. You are correct.

2. The polar equation r = 2*sin(theta) + 2*cos(theta) should be converted to the Cartesian equation x^2 + y^2 = 2y + 2x. You squared both sides of the equation correctly, but the simplification is not accurate yet. Let's continue the process:

r^2 = 2rsin(theta) + 2rcos(theta)
(x^2 + y^2) = 2y + 2x

x^2 + y^2 = 2y + 2x
x^2 - 2x + y^2 - 2y = 0
x^2 - 2x + 1 + y^2 - 2y + 1 = 2
(x^2 - 2x + 1) + (y^2 - 2y + 1) = 2 + 1 + 1
(x - 1)^2 + (y - 1)^2 = 4

The correct Cartesian equation is (x - 1)^2 + (y - 1)^2 = 4. Therefore, your answer is incorrect.

3. The polar equation r = tan(theta)sec(theta) is converted to r = sin(theta)/(cos(theta))^2. You have correctly simplified this equation to r^2(cos(theta))^2 = rsin(theta). However, we need to convert it to a Cartesian equation.

r^2(cos(theta))^2 = rsin(theta)
(x^2 + y^2)(x^2/(x^2 + y^2))^2 = y/(x^2 + y^2)

We can observe that both sides of the equation have a factor of (x^2 + y^2). Dividing both sides by (x^2 + y^2) gives:

x^2 = y

The correct Cartesian equation is x^2 = y. Therefore, your answer is correct.

To summarize:
1. Correct: x = 1
2. Incorrect: The correct Cartesian equation is (x - 1)^2 + (y - 1)^2 = 4
3. Correct: x^2 = y