Rewrite using $\cot\theta = \cos\theta/\sin\theta$:

$\dfrac{x\cdot\cos4x/\sin4x}{\sin^2x\cdot\cos^22x/\sin^22x}$

As $x\to0$: $\sin4x\approx4x$, $\sin2x\approx2x$, $\sin x\approx x$, all cosines $\to1$:

$\approx\dfrac{x\cdot\frac{1}{4x}}{x^2\cdot\frac{1}{4x^2}} = \dfrac{\frac{1}{4}}{\frac{1}{4}} = \mathbf{1}$