using the formula
x=rcosthetax=rcosθ
and
y=rsinthetay=rsinθ
substitute
(rsintheta)^2=((rcostheta-1)^2)/(rsintheta)-3(rcostheta)^2rsintheta(rsinθ)2=(rcosθ−1)2rsinθ−3(rcosθ)2rsinθ
expand brackets and simplify
r^2sin^2theta=(r^2cos^2theta-2rcostheta+1)/(rsintheta)-3r^2cos^2thetarsintheta*(rsintheta)/(rsintheta)r2sin2θ=r2cos2θ−2rcosθ+1rsinθ−3r2cos2θrsinθ⋅rsinθrsinθ
r^2sin^2theta=(r^2cos^2theta-2rcostheta+1)/(rsintheta)-(3r^2cos^2thetarsin^2theta)/(rsintheta)r2sin2θ=r2cos2θ−2rcosθ+1rsinθ−3r2cos2θrsin2θrsinθ
r^2sin^2theta=(r^2cos^2theta-2rcostheta+1-3r^2cos^2thetarsin^2theta)/(rsintheta)r2sin2θ=r2cos2θ−2rcosθ+1−3r2cos2θrsin2θrsinθ
simplify
r^2sin^2theta=(r^2cos^2theta-2rcostheta+1-3r^3cos^2thetasin^2theta)/(rsintheta)r2sin2θ=r2cos2θ−2rcosθ+1−3r3cos2θsin2θrsinθ
factorise
r^2sin^2theta=(r(rcos^2theta-2costheta-3r^2cos^2thetasin^2theta)+1)/(rsintheta)r2sin2θ=r(rcos2θ−2cosθ−3r2cos2θsin2θ)+1rsinθ
r^2sin^2theta=(cancel(r)(rcos^2theta-2costheta-3r^2cos^2thetasin^2theta))/(cancel(r)sintheta)+(1)/(rsintheta)
multiply by 1/sin^2theta
1/cancel(sin^2theta)*r^2cancel(sin^2theta)=1/sin^2theta((rcos^2theta-2costheta-3r^2cos^2thetasin^2theta)/(sintheta)+(1)/(rsintheta))
expand the brackets
r^2=(rcos^2theta-2costheta-3r^2cos^2thetasin^2theta)/(sin^3theta)+(1)/(rsin^3theta)
factorise
r^2=1/sin^3theta(rcos^2theta-2costheta-3r^2cos^2thetasin^2theta+1/r)
convert sin^2theta to 1-cos^2theta
r^2=1/sin^3theta(rcos^2theta-2costheta-3r^2cos^2theta(1-cos^2theta)+1/r)
expand the -3r^2cos^2theta(1-cos^2theta) brackets
r^2=1/sin^3theta(rcos^2theta-2costheta-3r^2cos^2theta+3r^2cos^4theta+1/r)
factorise the terms with rcos^2theta in it out
r^2=1/sin^3theta(rcos^2theta(1-3r+3rcos^2theta)-2costheta+1/r)
factorise
r^2=1/sin^3theta(costheta(rcostheta(1+3r(-1+cos^2theta)-2))+1/r)
