Find all angles $\phi$, if any, for which
the set $S=\left \{ \sin(\phi),\sin(2\phi),\sin(4\phi) \right \}$,
the set $C=\left \{ \cos(\phi),\cos(3\phi),\cos(9\phi) \right \}$,
and the set $T=\left \{ \tan(\phi),\tan(4\phi),\tan(16\phi) \right \}$ to be three equal sets.
What I have noticed; in $S$, the angles can be arranged to be in geometric progression. Similarly in $C$ and $T$, but I do not know how is that useful.
I could not even start solving this problem. Any help will be appreciated.
Just, consider three cases: $$\sin\theta=\cos\theta,$$
$$\sin\theta=\cos3\theta$$ and $$\sin\theta=\cos9\theta.$$ I got that we have no such value of $\theta$.