$\cos x+\cos 3x+\cos 5x+\cos 7x=0$, Any quick methods? How to solve the following equation by a quick method?

\begin{eqnarray}
\\\cos x+\cos 3x+\cos 5x+\cos 7x=0\\
\end{eqnarray}

If I normally solve the equation, it takes so long time for me.
I have typed it into a solution generator to see the steps. One of the steps shows that :

\begin{eqnarray}
\\\cos x+\cos 3x+\cos 5x+\cos 7x&=&0\\
\\-4\cos x+40\cos ^3x-96\cos ^5x+64\cos ^7x&=&0\\
\end{eqnarray}

How can I obtain this form? It seems very quick. or this quick methd do not exist?
Thank you for your attention

In case you want to crack the question through solving polynomial equation, let $cos(x)=y$
You've found $$64y^7-96y^5+10y^2-4y=0$$
It can be factorized to
$$4y(2y^2-1)(8y^4-8y^2+1)=0$$
Furthermore
$$2y^2-1=(\sqrt{2}y+1)(\sqrt{2}y-1)$$
and
$$(8y^4-8y^2+1)=(2\sqrt{2}y^2+1)^2-(8+4\sqrt{2})y^2=(2\sqrt{2}y^2+\sqrt{8+4\sqrt{2}}y+1)(2\sqrt{2}y^2-\sqrt{8+4\sqrt{2}}y+1)$$
Using quadratic formula you'll get the same solutions other people post.
