x=rcos(theta)x=rcos(θ)
y=rsin(theta)y=rsin(θ)
r^2=x^2+y^2r2=x2+y2
Make the necessary substitutions
3rsin(theta)=2(rcos(theta))^2-2rcos(theta)rsin(theta)-rcos(theta)3rsin(θ)=2(rcos(θ))2−2rcos(θ)rsin(θ)−rcos(θ)
Simplify
3rsin(theta)=2r^2cos^2(theta)-2r^2cos(theta)sin(theta)-rcos(theta)3rsin(θ)=2r2cos2(θ)−2r2cos(θ)sin(θ)−rcos(θ)
Add rcos(theta)rcos(θ) to both sides
3rsin(theta)+rcos(theta)=2r^2cos^2(theta)-2r^2cos(theta)sin(theta)3rsin(θ)+rcos(θ)=2r2cos2(θ)−2r2cos(θ)sin(θ)
Factor out rr and r^2r2
r(3sin(theta)+cos(theta))=r^2(2cos^2(theta)-2cos(theta)sin(theta))r(3sin(θ)+cos(θ))=r2(2cos2(θ)−2cos(θ)sin(θ))
Isolated r^2r2
(r(3sin(theta)+cos(theta)))/(2cos^2(theta)-2cos(theta)sin(theta))=(r^2cancel(2cos^2(theta)-2cos(theta)sin(theta)))/cancel(2cos^2(theta)-2cos(theta)sin(theta))
(r(3sin(theta)+cos(theta)))/(2cos^2(theta)-2cos(theta)sin(theta))=r^2
Gather r to the right hand side
((cancelr(3sin(theta)+cos(theta)))/(2cos^2(theta)-2cos(theta)sin(theta)))/cancelr=r^cancel2/cancelr
Simplify
(3sin(theta)+cos(theta))/(2cos^2(theta)-2cos(theta)sin(theta))=r
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