<s>[INST] $\cos x+\cos 3x+\cos 5x+\cos 7x=0$, 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
 [/INST] 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.
</s>