Don't worry about where the x^2 comes from, that is part of the answer and is a result of you solving the problem correctly.
You know that y=r\sin\theta by definition. Find y as a function of theta only.
Then you can eliminate theta when you find y as a function of x only.
I do not understand where you got this part from:
In polar coordinates,
x=r\cos\theta
and
y=r\sin\theta
Substitute your expression for r into these two equations. Solve the first one in such a way that you can plug your result into the second one to solve for y in terms of x.
At some point in class, you should have covered what the difference of two squares is. Just to remind you, it looks something like this:
a^2-b^2=(a+b)*(a-b)
I hope that helps.