Solving $\cos^2 x+\cos^2 2x +\cos^2 3x=1$ I picked this up from the IMO 1962.

Solve for $x$: 
  $$\cos^2 x+\cos^2 2x +\cos^2 3x=1$$ 

At first I thought it was trivial to bring this in an IMO, but I realized approaching it directly brings a power of 6 which is not all too friendly.
Is there a sneaky way to solve this?

Let $t:=\cos^2x$.
We have 
$$t+(2t-1)^2+(4t-3)^2t=1$$
or
$$16t^3-20t^2+6t=0.$$
The roots are $0,\dfrac12,\dfrac34$, nothing really difficult.
