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.
