Solving polynomials with compound angle formulae 

Solve $64x^6-96x^4+36x^2-3=0$ (Hint: Consider expanding $\cos 6\theta$ in terms of $\cos \theta$)


To solve this exercise I can simply derive an identity using de-Moivre theorem then shuffle the equation around. It turns out that substituting $x=\cos \theta$ reduces the equation to $\cos 6\theta = \frac{1}{2}$.
However, I'm wondering in the general case, is there any way to see this without the monster of a hint? In other words, given a polynomial, is there something I could look for in the coefficients that would make me think that deriving a compound angle formula may be a good idea?

It not always works in the general.
By the way, there are equations, which have nice real roots, but it not so easy to get these roots.
For example: $$x^5+x^4-12x^3-21x^2+x+5=0.$$
One of roots it's:

 $$2\left(\cos\frac{2\pi}{31}+\cos\frac{10\pi}{31}+\cos\frac{12\pi}{31}\right)$$

In another hand, there are equations, which we can solve by the similar idea, which you showed, but without any hint.
For example:
Solve the following equation.
$$8x(2x^2-1)(8x^4-8x^2+1)=1.$$
