<s>[INST] Function is small $o$ of $x^2$ I have to solve the following exercise:

Give $a$, $b$, $c \in \mathbb R$ such that 
  $$\frac{1}{1-\cos x} = \frac{a}{x^2} + b +cx^2 + o(x^2)$$
  for $x\to 0$.

Here's my attempt:
I know that $$\cos x = \frac{1}{2}\left( e^{ix} + e^{-ix}\right) = \frac{1}{2} \sum_{n=0}^\infty \frac{(ix)^{2n}}{(2n)!} = \frac{1}{2}\sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!} = \frac{1}{2}\left(1-\frac{x^2}{2}+o(x^2)\right)$$
for $x\to 0$.
The question now is how to resolve $\frac{1}{1-\cos x}$. This much looks like an application of the geometric series here. I assume I can use it since $\cos x < 1$ for all sufficiently small $x \neq 0$. I wouldn't know how to solve this exercise otherwise. Proceeding, it follows that
$$\frac{1}{1-\cos x} = \frac{1}{1-\left(\frac{1}{2} - \frac{x^2}{4} + o(x^2)\right)} = \sum_{n=0}^\infty\left(\frac{1}{2} - \frac{1}{4}x^2+o(x^2)\right)^n = 1+ \frac{1}{2}-\frac{1}{4}x^2+o(x^2).$$
Choosing $a=0$, $b = 1,5$ and $c=-\frac{1}{4}$ establishes the case.
 [/INST] hint
$$\cos(x)=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+o(x^6)$$ and after factoring out by $x^2$ , 
$$\frac{1}{1-X}=1+X+X^2+o(X^2).$$
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