Answer
$$\lim_{\theta\to0}\frac{\theta\cot4\theta}{\sin^2\theta\cot^22\theta}=1$$
Work Step by Step
$$A=\lim_{\theta\to0}\frac{\theta\cot4\theta}{\sin^2\theta\cot^22\theta}=\lim_{\theta\to0}\frac{\theta\times\frac{\cos4\theta}{\sin4\theta}}{\sin^2\theta\times\frac{\cos^22\theta}{\sin^22\theta}}$$
$$A=\lim_{\theta\to0}\frac{\theta\cos4\theta\sin^22\theta}{\sin^2\theta\sin4\theta\cos^22\theta}=\lim_{\theta\to0}\frac{\theta\cos4\theta\sin^22\theta}{\sin^2\theta(2\sin2\theta\cos2\theta)\cos^22\theta}$$
$$A=\lim_{\theta\to0}\frac{\theta\cos4\theta\sin2\theta}{2\sin^2\theta\cos^32\theta}$$
$$A=\lim_{\theta\to0}\frac{\theta\cos4\theta\times(2\sin\theta\cos\theta)}{2\sin^2\theta\cos^32\theta}=\lim_{\theta\to0}\frac{2\theta\cos4\theta\cos\theta}{2\sin\theta\cos^32\theta}$$
$$A=\lim_{\theta\to0}\frac{\theta\cos4\theta\cos\theta}{\sin\theta\cos^32\theta}$$
$$A=\lim_{\theta\to0}\frac{\theta}{\sin\theta}\times\lim_{\theta\to0}\frac{\cos4\theta\cos\theta}{\cos^32\theta}=X\times Y$$
1) Solve $X$:
$$X=\lim_{\theta\to0}\frac{\theta}{\sin\theta}=\Big(\lim_{\theta\to0}\frac{\sin\theta}{\theta}\Big)^{-1}$$
Apply Theorem 7 here:
$$X=(1)^{-1}=1$$
2) Solve $Y$:
$$Y=\lim_{\theta\to0}\frac{\cos4\theta\cos\theta}{\cos^32\theta}=\frac{\cos0\cos0}{\cos^30}=\frac{1\times1}{1^3}=1$$
Therefore, $$A=X\times Y=1\times1=1$$
