Let $f_n(x)=\sin^nx+\cos^nx$. For how many $x$ in $[0,\pi]$ is it true that $9f_4(x)-6f_6(x)=3f_2(x)$?

1. Writing $s=\sin^2x,c=\cos^2x$: $f_4=s^2+c^2=1-2sc$ and $f_6=s^3+c^3=1-3sc$.
2. $9(1-2sc)-6(1-3sc)=3-18sc+18sc=3=3f_2(x)$: an identity, true for every $x$.
3. So the equation holds for $\boxed{\text{all }x\in[0,\pi]}$ (infinitely many, i.e.\ ``more than $8$'').