use definition of derivative to find a derivative of a trig function

At this point, I can't use L'Hopital's rule. I tried to solve this using two different approaches and continue to get a 0 in the denominator even though I know that 0 should cancel out somehow. Here's the exercise:

Use the definition of the derivative to show that d/dx[cos x] = -sin x.

Hint: Use the limit laws in section 1.6 and the identity cos (A + B) = cos A cos B - sin A sin B.

Please help. This is one of two problems that I have left to do and have spent lots of time trying to solve to no avail.

Re: use definition of derivative to find a derivative of a trig function

Re: use definition of derivative to find a derivative of a trig function

Sorry, but I don't know what to use to cancel out the h's in the denominators. I've gone through the entire section 1.6 and don't see anything that I believe I can use to address this. Should I be looking more closely at the trigonometric identities or properties to resolve this?

Re: use definition of derivative to find a derivative of a trig function

you need two very important trig limits you've should have learned while covering limits ...

Re: use definition of derivative to find a derivative of a trig function

You can calculate these with L'Hôpital's rule, of course. That's what I do with when I come across it. But I've seen so many times that I just remember it.

- Hollywood