Explanation

(b)
$\lim_{x \rightarrow 0}\left( \cos x \right)^{\cot^{2}x} = \lim_{x \rightarrow 0}\left( 1 - 2\sin^{2}\frac{x}{2} \right)^{\cot^{2}x}$
$= e^{- \lim_{x \rightarrow 0}\ 2\sin^{2}{\frac{x}{2}\cot^{2}x}}$
$= e^{- \lim_{x \rightarrow 0}\ 2\sin^{2}{\frac{x}{2} \times \frac{\cos^{2}x}{4\sin^{2}{\frac{x}{2}\cos^{2}\frac{x}{2}}}}}$
$= e^{- \lim_{x \rightarrow 0}\ \frac{\cos^{2}x}{2\cos^{2}\frac{x}{2}}}$
$= e^{- \frac{1}{2}}$