$9.288 \operatorname{ctg} x-\operatorname{tg} x-2 \operatorname{tg} 2 x-4 \operatorname{tg} 4 x>8 \sqrt{3}$.
Solution. $\left(\frac{\cos x}{\sin x}-\frac{\sin x}{\cos x}\right)-2 \operatorname{tg} 2 x-4 \operatorname{tg} 4 x>8 \sqrt{3} \Leftrightarrow$

$\Leftrightarrow \frac{\cos 2 x}{2 \sin x \cdot \cos x}-\operatorname{tg} 2 x-2 \operatorname{tg} 4 x>4 \sqrt{3} \Leftrightarrow(\operatorname{ctg} 2 x-\operatorname{tg} 2 x)-2 \operatorname{tg} 4 x>4 \sqrt{3} \Leftrightarrow$

$\Leftrightarrow \operatorname{ctg} 4 x-\operatorname{tg} 4 x>2 \sqrt{3} \Leftrightarrow \operatorname{ctg} 8 x>\sqrt{3} \Leftrightarrow \pi n<8 x<\frac{\pi}{6}+\pi n$.

Answer: $x \in\left(\frac{\pi n}{8} ; \frac{\pi}{48}(6 n+1)\right)$.