We can understand the derivatives of the sine and cosine
functions
both algebraically and graphically. In the first video we take a
graphic look:

Algebraically

To get the derivatives algebraically, we need the two limits we
derived earlier:
$$\lim_{x \rightarrow 0} \frac{\sin(x)}{x} =1
\qquad \mbox{and} \qquad \lim_{x \rightarrow 0} \frac{\cos(x)
-1}{x} = 0,$$
as well as the trig addition identities: $$\begin{eqnarray*}\sin(A+B)
&=& \sin(A)\cos(B) + \cos(A)\sin(B), \cr
\cos(A+B) & = & \cos(A)\cos(B) - \sin(A)\sin(B).
\end{eqnarray*}$$