Answer
$$1$$
Work Step by Step
\begin{align*}
\lim _{\theta \rightarrow 0} \frac{\theta \cot 4 \theta}{\sin ^{2} \theta \cot ^{2} 2 \theta}&=\lim _{\theta \rightarrow 0} \frac{\theta^{\cos 4 \theta}}{\sin ^{2} \theta \frac{\cos ^{2} 2 \theta}{\sin ^{2}2 \theta}}\\
&=\lim _{\theta \rightarrow 0} \frac{\theta \cos 4 \theta \sin ^{2} 2 \theta}{\sin ^{2} \theta \cos ^{2} 2 \theta \sin 4 \theta}\\
&=\lim _{\theta \rightarrow 0} \frac{\theta \cos 4 \theta(2 \sin \theta \cos \theta)^{2}}{\sin ^{2} \theta \cos ^{2} 2 \theta \sin 4 \theta}\\
&=\lim _{\theta \rightarrow 0} \frac{\theta \cos 4 \theta\left(4 \sin ^{2} \theta \cos ^{2} \theta\right)}{\sin ^{2} \theta \cos ^{2} 2 \theta \sin 4 \theta}\\
&=\lim _{\theta \rightarrow 0} \frac{4 \theta \cos 4 \theta \cos ^{2} \theta}{\cos ^{2} 2 \theta \sin 4 \theta}\\
&=\lim _{\theta \rightarrow 0}\left(\frac{4 \theta}{\sin 4 \theta}\right)\left(\frac{\cos 4 \theta \cos ^{2} \theta}{\cos ^{2} 2 \theta}\right)\\
&=\lim _{\theta \rightarrow 0}\left(\frac{1}{\frac{\sin 4 \theta}{4 \theta}}\right)\left(\frac{\cos 4 \theta \cos ^{2} \theta}{\cos ^{2} 2 \theta}\right)\\
&=\left(\frac{1}{1}\right)\left(\frac{1-1^{2}}{1^{2}}\right)=1
\end{align*}
