Before we begin. Please notice that theta must not be a 2npi multiple of 0 or pi, because the cosecant function is undefined at the these points. This converts to the Cartesian restriction y != 0
Eventually, we will substitute tan^-1(y/x) for theta, therefore, we must look at the pristine function, r = (2cos(theta) - csc(theta))/(theta + 1), and see what happens at theta = pi/2 and (3pi)/2
r = -2/(pi + 2) and r = -2/(3pi + 2)
These are the y coordinates at x = 0.
y = -2/(pi + 2) and y = -2/(3pi + 2)
Now, we may proceed with the conversion.
For csc(theta):
y = rsin(theta)
1/sin(theta) = r/y
csc(theta) = r/y
csc(theta) = sqrt(x^2 + y^2)/y
r = (2cos(theta) - sqrt(x^2 + y^2)/y)/(theta + 1)
Multiply both sides by r:
r^2 = (2rcos(theta) - (x^2 + y^2)/y)/(theta + 1)
Substitute x for rcos(theta)
r^2 = (2x - (x^2 + y^2)/y)/(theta + 1)
Substitute x^2 + y^2 for r^2
(x^2 + y^2) = (2x - (x^2 + y^2)/y)/((theta + 1)
Multiply the right side by y/y:
(x^2 + y^2) = (2xy - x^2 - y^2)/(y(theta + 1))
The substitution for theta breaks the equation into 3 equations plus the 2 for x = 0 and y = 0:
Undefined for y = 0
y = -2/(pi + 2) and y = -2/(3pi + 2); x = 0
(x^2 + y^2) = (2xy - x^2 - y^2)/(y(tan^-1(y/x) + 1)); x > 0 and y > 0
(x^2 + y^2) = (2xy - x^2 - y^2)/(y(tan^-1(y/x) + pi + 1)); x < 0 and y != 0
(x^2 + y^2) = (2xy - x^2 - y^2)/(y(tan^-1(y/x) + 2pi + 1)); x > 0 and y < 0
