\begin{align*}
\frac{d}{dx}\sin x &= \lim_{\delta x \to 0} \frac{\sin(x + \delta x) - \sin x}{\delta x}\\
&= \lim_{\delta x \to 0} \frac{\sin x \cos \delta x + \cos x \sin \delta x - \sin x}{\delta x}\\
&= \lim_{\delta x \to 0} \frac{\sin x \cos \delta x - \sin x + \cos x \sin \delta x}{\delta x}\\
&= \lim_{\delta x \to 0} \frac{\sin x (\cos \delta x - 1) + \cos x \sin \delta x}{\delta x}\\
&= \lim_{\delta x \to 0} \frac{\sin x (\cos \delta x - 1)}{\delta x} + \lim_{\delta x \to 0} \frac{\cos x \sin \delta x}{\delta x}\\
&= \sin x \lim_{\delta x \to 0} \frac{\cos \delta x - 1}{\delta x} + \cos x \lim_{\delta x \to 0} \frac{\sin \delta x}{\delta x}\\
&= \sin x \times 0 + \cos x \times 1\\
\therefore \frac{d}{dx}\sin x &= \cos x\\
\end{align*}
Derivative of cos x
\begin{align*}
\frac{d}{dx}\cos x &= \lim_{\delta x \to 0} \frac{\cos (x + \delta x) - \cos x}{\delta x}\\
&= \lim_{\delta x \to 0} \frac{\cos x \cos \delta x - \sin x \sin \delta x - \cos x}{\delta x}\\
&= \lim_{\delta x \to 0} \frac{\cos x \cos \delta x - \cos x - \sin x \sin \delta x}{\delta x}\\
&= \lim_{\delta x \to 0} \frac{\cos x (\cos \delta x - 1) - \sin x \sin \delta x}{\delta x}\\
&= \lim_{\delta x \to 0} \frac{\cos x (\cos \delta x - 1)}{\delta x} - \lim_{\delta x \to 0} \frac{\sin x \sin \delta x}{\delta x}\\
&= \cos x \lim_{\delta x \to 0} \frac{\cos \delta x - 1}{\delta x} - \sin x \lim_{\delta x \to 0} \frac{\sin \delta x}{\delta x}\\
&= \cos x \times 0 - \sin x \times 1\\
\therefore \frac{d}{dx}\cos x&= - \sin x
\end{align*}