Differentiating y=Cos x by limit definition (First year degree level calculus topic)

Hey all,
I've just come to a question for Calculus 1 (at Uni) which I'm drawing blanks with.

Basically I have to differentiate:

$y = Cos x$

Via limit definition (obviously $dy/dx = - Sin x$, although that's what we have to prove/show via limit definition)

In the question we're also given the following:

$\lim_{x\to 0} \frac{Sin x}{x} = 1$

And a "may or may not be useful, depending on your method":

$\lim_{x\to 0} \frac{(Cos x - 1)}{x} = 1$

Any pointers would be greatly appreciated

I tried plugging $y = Cos x$ into the limit/derivative formula, but ended up drawing a blank because we'd end up looking for $\lim_{h\to 0} h = 0$ in the denominator, which is an obvious pitfall.

And I can't see of a way of using the Substitution Law (albeit I'm not too confident on that law yet)

Thanks
Tristan

Are you sure that your second given limit is equal to 1? I'm fairly sure it should be 0