Question

$\underset{x\to a}{\lim }\frac{\cos x-\cos a}{x-a}$

Answer 1

$$\begin{gathered} \underset{x\to a}{\lim }\frac{\cos x-\cos a}{x-a} \\ =\underset{x\to a}{\lim }\frac{-2\sin \left({\displaystyle \frac{x+a}{2}}\right)\sin \left({\displaystyle \frac{x-a}{2}}\right)}{2\left({\displaystyle \frac{x-a}{2}}\right)}\left[\because \cos \mathrm{A}-\cos \mathrm{B}-2\sin \left(\frac{\mathrm{A}-\mathrm{B}}{2}\right)\sin \left(\frac{\mathrm{A}+\mathrm{B}}{2}\right)\right] \\ =\underset{x\to a}{\lim }-\sin \left(\frac{x+a}{2}\right)\left[\because \underset{\mathrm{\theta }\to a}{\lim }\sin \frac{\left(\mathrm{\theta }-a\right)}{\left(\mathrm{\theta }-a\right)}=1\right] \\ =-\sin \left(\frac{2a}{2}\right) \\ \Rightarrow -\sin a \end{gathered}$$