8.219. $3(\operatorname{ctg} t-\operatorname{tg} t)+4 \sin 2 t=0$.

8.219. $3(\cot t-\tan t)+4 \sin 2 t=0$.
## Solution.

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

From the condition we have:

$$
\begin{aligned}
& 3(\operatorname{ctg} t-\operatorname{tg} t)+4 \sin 2 t=0 \Leftrightarrow 3 \cdot\left(\frac{\cos t}{\sin t}-\frac{\sin t}{\cos t}\right)+4 \sin 2 t=0 \Leftrightarrow \\
& \Leftrightarrow 3 \cdot \frac{\cos ^{2} t-\sin ^{2} t}{\sin t \cos t}+4 \sin 2 t=0 \Leftrightarrow \frac{6 \cos 2 t}{\sin 2 t}+4 \sin 2 t=0 \Leftrightarrow \\
& \Leftrightarrow 3 \cos 2 t+2 \sin ^{2} 2 t=0 \Leftrightarrow 3 \cos 2 t+2\left(1-\cos ^{2} 2 t\right)=0 \Leftrightarrow \\
& \Leftrightarrow 2 \cos ^{2} 2 t-3 \cos 2 t-2=0
\end{aligned}
$$

Solving this equation as a quadratic in $\cos 2 t$, we find $\cos 2 t=2, \varnothing ; \cos 2 t=-\frac{1}{2}, 2 t= \pm \frac{2}{3} \pi+2 \pi n, t= \pm \frac{\pi}{3}+\pi n=\frac{\pi}{3}(3 n \pm 1)$, $n \in Z$.

Answer: $t=\frac{\pi}{3}(3 n \pm 1), n \in Z$.