lim x tends to a [ (cos x - cos a)/ (cot x - cot a)] =?

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{L}=\underset{\mathrm{x}\to \mathrm{a}}{\lim } \left(\frac{\mathrm{cosx}-\mathrm{cosa}}{\mathrm{cotx}-\mathrm{cota}}\right) \\ \mathrm{Using} \mathrm{identity} \mathrm{cosC}-\mathrm{cosD}=2\sin \left(\frac{\mathrm{C}+\mathrm{D}}{2}\right)\sin \left(\frac{\mathrm{D}-\mathrm{C}}{2}\right), \mathrm{in} \mathrm{numerator} \mathrm{we} \mathrm{get}: \\ \mathrm{L}=\underset{\mathrm{x}\to \mathrm{a}}{\lim }\left(\frac{2\sin \left({\displaystyle \frac{\mathrm{x}+\mathrm{a}}{2}}\right)\sin \left({\displaystyle \frac{\mathrm{a}-\mathrm{x}}{2}}\right)}{{\displaystyle \frac{\mathrm{cosx}}{\mathrm{sinx}}}-{\displaystyle \frac{\mathrm{cosa}}{\mathrm{sina}}}}\right) \\ =\underset{\mathrm{x}\to \mathrm{a}}{\lim }\left(\frac{2\sin \left({\displaystyle \frac{\mathrm{x}+\mathrm{a}}{2}}\right)\sin \left({\displaystyle \frac{\mathrm{a}-\mathrm{x}}{2}}\right)}{{\displaystyle \frac{\mathrm{sina}.\mathrm{cosx}-\mathrm{cosa}.\mathrm{sinx}}{\mathrm{sinx}.\mathrm{sina}}}}\right) \\ =\underset{\mathrm{x}\to \mathrm{a}}{\lim }2\mathrm{sinx}.\mathrm{sina}\left[\frac{2\sin \left({\displaystyle \frac{\mathrm{x}+\mathrm{a}}{2}}\right)\sin \left({\displaystyle \frac{\mathrm{a}-\mathrm{x}}{2}}\right)}{{\displaystyle \sin \left(\mathrm{a}-\mathrm{x}\right)}}\right] \\ =\underset{\mathrm{x}\to \mathrm{a}}{\lim }2\mathrm{sinx}.\mathrm{sina}\left[\frac{2\sin \left({\displaystyle \frac{\mathrm{x}+\mathrm{a}}{2}}\right)\sin \left({\displaystyle \frac{\mathrm{a}-\mathrm{x}}{2}}\right)}{{\displaystyle 2\sin \left(\frac{\mathrm{a}-\mathrm{x}}{2}\right)\cos \left(\frac{\mathrm{a}-\mathrm{x}}{2}\right)}}\right] \left[\mathrm{sinA}=2\sin \left(\frac{\mathrm{A}}{2}\right)\cos \left(\frac{\mathrm{A}}{2}\right)\right] \\ =\underset{\mathrm{x}\to \mathrm{a}}{\lim } \mathrm{sinx}.\mathrm{sina}\left[\frac{\sin \left({\displaystyle \frac{\mathrm{x}+\mathrm{a}}{2}}\right)}{{\displaystyle \cos \left(\frac{\mathrm{a}-\mathrm{x}}{2}\right)}}\right] \\ \mathrm{Applying} \mathrm{limit} \mathrm{we} \mathrm{get}: \\ \mathrm{L}=\mathrm{sina}.\mathrm{sina}\left[\frac{\sin \left({\displaystyle \frac{\mathrm{a}+\mathrm{a}}{2}}\right)}{{\displaystyle \cos \left(\frac{\mathrm{a}-\mathrm{a}}{2}\right)}}\right] \\ =\mathrm{sina}.\mathrm{sina}\left[\frac{\mathrm{sina}}{{\displaystyle \cos \left(0\right)}}\right] \\ ={\sin }^3\left(\mathrm{a}\right) \left[\mathrm{As} \cos \left(0\right)=1\right] \\ \Rightarrow \underset{\mathbf{x}\mathbf{\to }\mathbf{a}}{\mathbf{lim}}\mathbf{ }\left(\frac{\mathbf{cosx}\mathbf{-}\mathbf{cosa}}{\mathbf{cotx}\mathbf{-}\mathbf{cota}}\right)\mathbf{=}{\mathbf{sin}}^{\mathbf{3}}\left(\mathbf{a}\right) \end{gathered}$$

Hope this information will clear your doubts about this topic.

If you have any doubts just ask here on the ask and answer forum and our experts will try to help you out as soon as possible.
Regards