Explanation

(c)
$\lim_{\alpha \rightarrow 0}\frac{\left\lbrack \begin{array}{r}
cosec^{- 1}\left( \sec\alpha \right) + \cot^{- 1}{\left( \tan\alpha \right) +} \\
\cot^{- 1}{\cos\left( \sin^{- 1}\alpha \right)}
\end{array} \right\rbrack}{\alpha}$
$= \lim_{\alpha \rightarrow 0}\$
$\frac{\left\lbrack \begin{array}{r}
cosec^{- 1}\left( \ {cosec}\left( \frac{\pi}{2} - \alpha \right) \right) + \cot^{- 1}\left( \cot\left( \frac{\pi}{2} - \alpha \right) \right) \\
+ \cot^{- 1}{\cos{\lbrack\cos^{- 1}\sqrt{(1 - \alpha^{2}})\rbrack}}
\end{array} \right\rbrack}{\alpha}$
$= \lim_{\alpha \rightarrow 0}\frac{\pi - 2\alpha + \cot^{- 1}\sqrt{1 - \alpha^{2}}}{\alpha}$
$= \lim_{\alpha \rightarrow 0}\ \frac{- 2 - \frac{1}{1 + 1 - \alpha^{2}}\left( \frac{1}{2\sqrt{1 - \alpha^{2}}}( - 2\alpha) \right)}{1}$
$= - 2$ [by L’Hospital’s rule]