<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] we can use product to sum  formula. we have 
$$\begin{align}(1- 2\cos 3x)(-\cos 2x - \cos x) &= 2\cos 3x \cos 2x + 2 \cos 3x \cos x - \cos 2x - \cos x\\
 &= (\cos 5x + \cos x) + (\cos 4x + \cos 2x) - \cos 2x - \cos x\\
&=\cos 5x+\cos 4x\end{align}$$
dividing it out by $1- 2\cos 3x$ gives the result.
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