How to factorize $\dfrac{\cos(3x)-\cos(x)}{\tan(2x)-\tan(x)}$?
Which trigonometric identities to use?
I'm stuck when it comes to $\tan(2x)+\tan(x)$. I don't know which identity to use to turn it into the product.
I was thinking of just transforming them to sines and cosines, but also doesn't get me anywhere.
Thanks for help in advance.
$$\frac{\cos3x-\cos x}{\tan2x-\tan x}$$ Sum-to-product rule on top, decompose $\tan$ into $\sin/\cos$ on bottom: $$=\frac{-2\sin2x\sin x}{\sin 2x/\cos2x-\sin x/\cos x}$$ Expand $\sin2x$ using double-angle formula: $$=\frac{-2(2\sin x\cos x)\sin x}{(2\sin x\cos x)/\cos2x-\sin x/\cos x}$$ Cancel $\sin x$ and rewrite $\cos2x$ using double-angle formula: $$=\frac{-4\sin x\cos x}{(2\cos x)/(2\cos^2x-1)-1/\cos x}$$ Multiply halves by $\cos x$ and move the $-1$ in the denominator onto the fraction to its left: $$=\frac{-4\sin x\cos^2x}{(2\cos^2x-(2\cos^2x-1)/(2\cos^2x-1)}$$ The denominator simplifies to $1/(2\cos^2x-1)$: $$=-4\sin x\cos^2x(2\cos^2x-1)$$ Collect like terms by reverse double-angle formulas: $$=-2\cos x\sin2x\cos2x=-\cos x\sin4x$$
First note the following identities: \begin{align} \cos (3x) &= 4 \cos ^3 (x) - 3 \cos (x) \\ \sin(2x) &= 2 \sin(x) \cos(x) \\ \cos(2x) &= 2 \cos^2(x) - 1 \end{align} Substituting into the expression yields \begin{multline} \frac{\cos(3x) - \cos(x)}{\tan(2x) - \tan(x)} = \frac{4 \cos(x)(\cos^2(x) - 1)}{\frac{2 \sin(x) \cos(x)}{2 \cos^2(x) - 1} - \frac{\sin(x)}{\cos(x)}} = \frac{4 \cos^2(x)(\cos^2(x) - 1)(2\cos^2(x) - 1)}{2 \sin(x) \cos^2(x) - \sin(x)(2\cos^2(x) - 1)} \\= 4 \frac{\cos^2(x)}{\sin(x)}(\cos^2(x) - 1)(2\cos^2(x) - 1). \end{multline} Now, substituting back in $2 \cos^2(x) - 1=\cos(2x)$ and $\cos^2(x) - 1 = -\sin^2(x)$ yields \begin{multline}4 \frac{\cos^2(x)}{\sin(x)}(\cos^2(x) - 1)(2\cos^2(x) - 1) = -4 \cos^2(x) \sin(x) \cos(2x)\\ = -2\cos(x)\sin(2x)\cos(2x) = -\cos(x)\sin(4x)\end{multline} where in the last two equalities I have used $2 \sin(x) \cos(x) = \sin(2x)$.
Hint:
$$\tan2x-\tan x=\cdots=\dfrac{\sin(2x -x)}{\cos x\cos2x}$$
Using Prosthaphaeresis Formulas, $$\cos3x-\cos x=2\sin\dfrac{3x+x}2\sin\dfrac{x-3x}2$$
Finally, $\sin(-y)=-\sin y$.
