limits - Evaluating $lim_{xto 0} ; frac{2cos(a+x)-cos(a+2x)-cos(a)}{x^2}$

So I am asked to find this limit

$$\lim_{x\to 0} \; \frac{2\cos(a+x)-\cos(a+2x)-\cos(a)}{x^2}$$

I know I am supped to use the trig identity $\cos(u+v)=\cos(u)\cos(v)-\sin(u)\sin(v)$ but I am having trouble with the denominator. I am trying to use the common limit $\lim_{x \to 0} \frac{sin(x)}{x}$ but if I expand out every term, I don't have a sin everywhere. How would I deal with that? I tried simplifying the answer but I am honesty getting nowhere.

Could I have some help with how I would simplify? Thanks.

Answer

HINT:

Using Prosthaphaeresis Formula,
$$\cos(a+2x)+\cos a=2\cos(a+x)\cos x$$

Now $\dfrac{1-\cos x}{x^2}=\left(\dfrac{\sin x}x\right)^2\cdot\dfrac1{1+\cos x}$