$5.5 \sin 3 x+\sin 5 x=\sin 4 x$.
5.5 Applying the formula for the sum of sines (4.19), after transformations we get
$2 \sin 4 x \cos x=\sin 4 x ; \sin 4 x(2 \cos x-1)=0$.

Thus:

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

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