Prove $\frac{\cos 5x + \cos 4x}{1-2\cos 3x} = -\cos 2x -\cos x$

Question

Prove that $$\frac{\cos 5x + \cos 4x}{1-2\cos 3x} = -\cos 2x -\cos x$$

I have tried applying transformation formula on the numerator and I am stuck as it leads me nowhere to the answer.

Please help me prove this. Thank You!

Answer 1

Note

\begin{align} &\frac{\cos 5x + \cos 4x}{1-2\cos 3x} \\ =& \frac{(\cos 5x +\cos x) -\cos x+ (\cos 4x+ \cos2x)-\cos2x}{1-2\cos 3x} \\ =& \frac{2\cos 3x\cos 2x -\cos x+ 2\cos 3x \cos x-\cos2x}{1-2\cos 3x} \\ =& \frac{(2\cos 3x-1)(\cos 2x+\cos x)}{1-2\cos 3x} \\ =&-\cos 2x -\cos x \end{align}

Answer 2

Hint:

Use Prosthaphaeresis Formulas on

$$\cos5x+\cos4x, \cos2x+\cos x$$

For $\cos\dfrac{3x}2\ne0,$

$$2\cos3x-1=2\left(2\cos^2\dfrac{3x}2-1\right)-1=\dfrac{\cos\dfrac{9x}2}{\cos\dfrac{3x}2}$$

Answer 3

We will use well known formula: $\cos(a)+\cos(b) = 2\cos\frac{a-b}{2}\cos\frac{a-b}{2}. \\$

$$\frac{\cos5x+\cos4x}{1-2\cos3x}=-\cos2x-\cos x \\$$

$$\iff \cos5x+\cos4x=-\cos2x - \cos x+2\cos2x\cos3x+2\cos x \cos3x \\ $$

$$\iff \cos5x+\cos x + \cos4x + \cos2x = 2\cos2x\cos3x+2\cos x \cos3x \\ $$

$$\iff 2\cos2x\cos3x+2\cos x \cos3x = 2\cos2x\cos3x+2\cos x \cos3x$$