<s>[INST] Show that $ \frac{\cos5x + \cos4x} {1-2\cos3x} = -\cos2x -\cos x $ The Question reads - 
$$ \frac{\cos5x + \cos4x} {1-2\cos3x} = -\cos2x -\cos x $$ 
I tried using the obvious approach by converting $5x , 4x $ and $ 3x$ to either $2x$ or $x$ but all that seemed to do was to further complicate the fraction. Any hints would be much appreciated.
I also tried applying various identities but to no avail.
 [/INST] Let's apply the identity $\cos (x+y)+\cos(x-y)=2\cos x \cos y$ to the product 
$$(1-2\cos 3x)(-\cos 2x-\cos x)=-\cos 2x -\cos x+2\cos 3x \cos 2x+2\cos 3x \cos x$$
Note that $2\cos 3x \cos 2x=\cos x+\cos 5x$ while $2\cos 3x \cos x=\cos 2x+\cos 4x$
Thus
$$\begin{align}
(1-2\cos 3x)(-\cos 2x-\cos x)&=-\cos 2x -\cos x+2\cos 3x \cos 2x+2\cos 3x \cos x\\\\
&=-\cos 2x -\cos x+(\cos x+\cos 5x)+(\cos 2x +\cos 4x)\\\\
&=\cos 5x + \cos 4x
\end{align}$$
which was to be shown!
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