8.003. $\frac{4 \operatorname{ctg} x}{1+\operatorname{ctg}^{2} x}+\sin ^{2} 2 x+1=0$.

8.003. $\frac{4 \cot x}{1+\cot^{2} x}+\sin ^{2} 2 x+1=0$.
## Solution.

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

## From the condition

$$
\begin{aligned}
& \frac{\frac{4 \cos x}{\sin x}}{1+\frac{\cos ^{2} x}{\sin ^{2} x}}+\sin ^{2} 2 x+1=0, \Leftrightarrow \sin ^{2} 2 x+2 \sin 2 x+1=0 \Leftrightarrow \\
& \Leftrightarrow(\sin 2 x+1)^{2}=0, \sin 2 x=-1
\end{aligned}
$$

Then $2 x=-\frac{\pi}{2}+2 \pi k, k \in Z, x=-\frac{\pi}{4}+\pi k=\frac{\pi}{4}(4 k-1), \quad k \in Z$.

Answer: $\quad x=\frac{\pi}{4}(4 k-1), k \in Z$.