Use the definition of derivative to find d/dx cos(x).
by the limit definition of the derivative:
$\displaystyle \frac d{dx}f(x) = \lim_{h \to 0} \frac {f(x + h) - f(x)}h$
thus, $\displaystyle \frac d{dx} \cos(x) = \lim_{h \to 0}\frac {\cos (x + h) - \cos x }h$
now, continue (begin by using the addition formula for cosine to expand $\displaystyle \cos (x + h)$ in the numerator
You can use the addtion formula for cos.
$\displaystyle \frac{d}{dx}[cos(x)]=\lim_{h\rightarrow{0}}\frac{cos(x+h)-cos(x)}{h}$
=$\displaystyle \lim_{h\rightarrow{0}}\frac{cos(x)cos(h)-sin(x)sin(h)-cos(x)}{h}$
=$\displaystyle \lim_{h\rightarrow{0}}\left[cos(x)\left(\frac{cos(h)-1}{h}\right)-sin(x)\left(\frac{sin(h)}{h}\right)\right]$
Can you finish?.