$\lim_{x \to 0} (\cos x)^{\cot x}$

Question

The following question is from cengage calculus . Illustration 2.95 but the explanation isn't clear to me

$\lim_{x \to 0} (\cos x)^{\cot x}$

It is to be solved by using the identity : $\lim_{x \to 0} (1+x)^{\frac{1}{x}} = e$

Answer 1

$$\begin{align} \lim_{x \to 0} (\cos x)^{\cot x} & =\lim_{x \to 0} ((1+(\cos x-1))^{\frac{1}{\cos x -1}})^{(\cos x -1)\cot x}\\\\ & =\lim_{x \to 0} e^{(\cos x -1)\cot x}\\\\ & =e^{{\lim_{x \to 0} {(\cos x -1)\cot x}}}\\\\& =e^0\\\\ &=1 \end{align}$$