Given: rsin(theta) = 2theta+cot(theta)-tan(theta)
Substitute theta = tan^-1(y/x)
rsin(theta) = 2tan^-1(y/x)+cot(theta)-tan(theta)
Substitute cot(theta) = cos(theta)/sin(theta)
rsin(theta) = 2tan^-1(y/x)+cos(theta)/sin(theta)-tan(theta)
Substitute tan(theta) = sin(theta)/cos(theta)
rsin(theta) = 2tan^-1(y/x)+cos(theta)/sin(theta)-sin(theta)/cos(theta)
Multiply the last 2 terms by in the form of r/r:
rsin(theta) = 2tan^-1(y/x)+(rcos(theta))/(rsin(theta))-(rsin(theta))/(rcos(theta))
Substitute rsin(theta) = y:
y = 2tan^-1(y/x)+(rcos(theta))/y-y/(rcos(theta))
Substitute rcos(theta) = x:
y = 2tan^-1(y/x)+x/y-y/x
Done.
