$\cos x+\cos 3x+\cos 5x+\cos 7x=0$, 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

Use product identity $2\cos(x)\cos (y)=\cos(x+y)+\cos(x-y)$
So $\cos x+\cos7x=2\cos4x\cos3x$
And $\cos3x+\cos5x=2\cos4x\cos x$.
Factoring out $2\cos4x$, we get $2\cos4x (\cos3x+\cos x)=0$.
You can solve the 1st factor now right? For the second factor, use the above identity again and you will be done.:)
