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$$ \begin{align} (\sin x)' &= \lim_{\Delta x \to 0} \frac{\sin(x+\Delta x) - \sin x}{\Delta x} \\ &= \lim_{\Delta x \to 0} \frac{2\sin \frac{(x+\Delta x)-x}{2} \cos \frac{(x+\Delta x)+x}{2}}{\Delta x} \\ &= \lim_{\Delta x \to 0} \frac{\sin \frac{\Delta x}{2} \cos \frac{2x+\Delta x}{2}}{\Delta x/2} \\ &= \lim_{\Delta x \to 0} \cos \frac{2x+\Delta x}{2} &\qquad (\frac{\sin x}{x} \sim 1) \\ &= \cos x \end{align} $$