8.201. $\sin 2 x+2 \operatorname{ctg} x=3$.

8.201. $\sin 2 x+2 \cot x=3$.
## Solution.

Domain of definition: $\sin x \neq 0$.

Since $\sin 2 \alpha=\frac{2 \operatorname{tg} \alpha}{1+\operatorname{tg}^{2} \alpha}$, we have

$\frac{2 \operatorname{tg} x}{1+\operatorname{tg}^{2} x}+\frac{2}{\operatorname{tg} x}-3=0 \Rightarrow 3 \operatorname{tg}^{3} x-4 \operatorname{tg}^{2} x+3 \operatorname{tg} x-2=0$.

Let $\operatorname{tg} x=y$, then

$$
\begin{aligned}
& 3 y^{3}-4 y^{2}+3 y-2=0, \quad 3 y^{3}-3 y^{2}-y^{2}+2 y+y-1-1=0 \\
& 3 y^{2}(y-1)-(y-1)^{2}+(y-1)=0,(y-1)\left(3 y^{2}-y+2\right)=0
\end{aligned}
$$

## From this

1) $y-1=0, y=1, \operatorname{tg} x=1, x=\frac{\pi}{4}+\pi n=\frac{\pi}{4}(4 n+1), n \in Z$;
2) $3 y^{2}-y+2 \neq 0(D<0), \varnothing$.

Answer: $x=\frac{\pi}{4}(4 n+1), n \in Z$.