8.165. $2 \tan x - 2 \cot x = 3$.
Solution.

Domain of definition: $\left\{\begin{array}{l}\cos x \neq 0 \\ \sin x \neq 0\end{array}\right.$

We have

$2\left(\frac{\sin x}{\cos x}-\frac{\cos x}{\sin x}\right)=3, \frac{\sin ^{2} x-\cos ^{2} x}{\sin x \cos x}=\frac{3}{2} \Leftrightarrow \frac{\cos 2 x}{\sin 2 x}=-\frac{3}{4}$, $\operatorname{ctg} 2 x=-\frac{3}{4}, \quad 2 x=-\operatorname{arcctg} \frac{3}{4}+\pi n, \quad x=-\frac{1}{2} \operatorname{arcctg} \frac{3}{4}+\frac{\pi n}{2}, n \in Z$.

Answer: $\quad x=-\frac{1}{2} \operatorname{arcctg} \frac{3}{4}+\frac{\pi n}{2}, n \in Z$.