Finding $\phi$ so that we have $3$ equal sets 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 $\theta$.
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 $\theta$.