8.129. $\sin x-\sin 3 x-\sin 5 x+\sin 7 x=0$.

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8.129. $\sin x-\sin 3 x-\sin 5 x+\sin 7 x=0$.
Solution.

We have

$$
\begin{aligned}
& (\sin x+\sin 7 x)-(\sin 3 x+\sin 5 x)=0 \Leftrightarrow 2 \sin \frac{x+7 x}{2} \cos \frac{x-7 x}{2}- \\
& -2 \sin \frac{3 x+5 x}{2} \cos \frac{3 x-5 x}{2}=0,2 \sin 4 x \cos 3 x-2 \sin 4 x \cos x=0 \\
& 2 \sin 4 x(\cos 3 x-\cos x)=0
\end{aligned}
$$

From this:

1) $\sin 4 x=0,4 x=\pi k, \quad x_{1}=\frac{\pi k}{4}, k \in Z$;
2) $\cos 3 x-\cos x=0 \Leftrightarrow 2 \sin \frac{3 x+x}{2} \sin \frac{x-3 x}{2}=0,-\sin 2 x \sin x=0$, from which

a) $\sin 2 x=0,2 x=\pi n, x_{2}=\frac{\pi n}{2}, n \in Z$

b) $\sin x=0, x_{3}=\pi l, \quad l \in Z ; x_{2}$ and $x_{3}$ are included in $x_{1}$.

Answer: $x=\frac{\pi k}{4}, k \in Z$.