Evaluate the following limits:

$=\lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(x-a)}$

$=\lim _{x \rightarrow a} \frac{\left(2 \times \cos \frac{x+a}{2} \sin \frac{x-a}{2}\right)}{(x-a)}\left[\because \sin x-\sin a=2 \times \cos \frac{x+a}{2} \sin \frac{x-a}{2}\right]$

$=1 \times \lim _{x \rightarrow a} \cos \frac{x+a}{2}\left[\because \lim _{x \rightarrow a} \frac{\sin \theta}{\theta}=1\right]$

$=\cos \frac{a+a}{2}$

$\therefore \lim _{x \rightarrow a} \frac{\sin x-\sin a}{x-a}=\cos a$