Evaluate the following limits:

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

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

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

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

$=-1 \times \sin \frac{(a+a)}{2}$

$=-1 \times \sin \frac{2 a}{2}$

$=-\sin (a)$

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