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