Originally Posted by
Jhevon by defintion
$\displaystyle f'(a) = \lim_{x \to a} \frac {f(x) - f(a)}{x - a}$
here you have $\displaystyle f(x) = \cos x$
so you can just say it is the derivative of cos(a) and just state that as the limit, namely, -sin(a)
if you would rather calculate it manually, note that an alternate definition for the derivative is as follows
$\displaystyle f'(x) = \lim_{h \to 0} \frac {f(x + h) - f(x)}h$
it is easier to find this particular limit in that form
getting your limit in that form is not as hard as you think
let $\displaystyle x = a + h$, then $\displaystyle \lim_{x \to a} \frac {\cos x - \cos a}{x - a}$ becomes $\displaystyle \lim_{h \to 0} \frac { \cos (a + h) - \cos a}h$
now proceed using the addition formula for cosine to simplify
(note, you will need your special trig limits here)