8.096. $\sin 3 x+\sin 5 x=\sin 4 x$.

8.096. $\sin 3 x+\sin 5 x=\sin 4 x$.
Solution.

Rewrite the given equation as

$$
\begin{aligned}
& 2 \sin \frac{3 x+5 x}{2} \cos \frac{3 x-5 x}{2}-\sin 4 x=0,2 \sin 4 x \cos x-\sin 4 x=0 \\
& \sin 4 x(2 \cos x-1)=0
\end{aligned}
$$

From this:

1) $\sin 4 x=0,4 x=\pi n, x_{1}=\frac{\pi n}{4}, n \in Z$;
2) $2 \cos x-1=0, \quad \cos x=\frac{1}{2}, \quad x_{2}= \pm \frac{\pi}{3}+2 \pi k=\frac{\pi}{3}(6 k \pm 1) \quad k \in Z$.

Answer: $\quad x_{1}=\frac{\pi n}{4} ; x_{2}=\frac{\pi}{3}(6 k \pm 1), n, k \in Z$.