2. [5 points] Solve the equation $\cos 9 x-\cos 5 x-\sqrt{2} \cos 4 x+\sin 9 x+\sin 5 x=0$.
Answer: $x=\frac{\pi}{8}+\frac{\pi k}{2}, x=\frac{\pi}{20}+\frac{2 \pi k}{5}, x=\frac{\pi}{12}+\frac{2 \pi k}{9}, k \in \mathbb{Z}$.

Solution. The given equation is equivalent to the following:

$-2 \sin 2 x \sin 7 x+2 \sin 7 x \cos 2 x=\sqrt{2} \cos 4 x \Leftrightarrow 2 \sin 7 x(\cos 2 x-\sin 2 x)=\sqrt{2}\left(\cos ^{2} 2 x-\sin ^{2} 2 x\right) \Leftrightarrow$

![](https://cdn.mathpix.com/cropped/2024_05_06_ffd4340bf000b94108d2g-13.jpg?height=276&width=1670&top_left_y=1324&top_left_x=273)

Finally, we get $x=\frac{\pi}{8}+\frac{\pi k}{2}, x=\frac{\pi}{20}+\frac{2 \pi k}{5}, x=\frac{\pi}{12}+\frac{2 \pi k}{9}, k \in \mathbb{Z}$.