8.044. $\sqrt{3} \sin 2 x+\cos 5 x-\cos 9 x=0$.

8.044. $\sqrt{3} \sin 2 x+\cos 5 x-\cos 9 x=0$.
Solution.

Rewrite the given equation as

$$
\begin{aligned}
& \sqrt{3} \sin 2 x-2 \sin \frac{5 x+9 x}{2} \sin \frac{5 x-9 x}{2}=0 \Leftrightarrow \\
& \Leftrightarrow \sqrt{3} \sin 2 x+2 \sin 7 x \sin 2 x=0 \Leftrightarrow \sin 2 x(\sqrt{3}+2 \sin 7 x)=0
\end{aligned}
$$

From this:

1) $\sin 2 x=0, 2 x=\pi n, x_{1}=\frac{\pi n}{2}, n \in Z$;
2) $\sqrt{3}+2 \sin 7 x=0, \sin 7 x=-\frac{\sqrt{3}}{2}, 7 x=(-1)^{k+1} \frac{\pi}{3}+\pi k$, $x_{2}=(-1)^{k+1} \frac{\pi}{21}+\frac{\pi k}{7}, k \in Z$

Answer: $x_{1}=\frac{\pi n}{2} ; x_{2}=(-1)^{k+1} \frac{\pi}{21}+\frac{\pi k}{7}, n, k \in Z$