What Regular Polygon Should I Draw?

A solution using roots of unity: first note that the expression we want is $S=\csc^2(\frac{2\pi}{9})+\csc^2(\frac{4\pi}{9})+\csc^2(\frac{8\pi}{9})$ . Now consider the 9th cyclotomic polynomial $\Phi_9=x^6+x^3+1$ which has roots $x=\text{cis}(\frac{2\pi k}{9})$ for integer $k$ with $\gcd(k,9)=1$ . Then with the transformation $t=x+\frac{1}{x}=2\cos(\frac{2\pi k}{9})$ we obtain the polynomial $t^3-3t+1$ whose roots are $t=2\cos(\frac{2\pi k}{9})$ for $k \in \{1,2,4\}$ . Finally make $y=\frac{4}{4-t^2} \implies t^2=\frac{4(y-1)}{y}$ to obtain the polynomial $3y^3-36y^2+96y-64$ whose roots are $y=\csc^2(\frac{2\pi k}{9})$ for $k \in \{1,2,4\}$ . By Vieta's formulas, $S=-\frac{-36}{3}=\boxed{12}$ .

I know there's a lot of manipulation involved in this method and its quite lengthy. But I thought this would be another solution. It was a nice problem!