2. [5 points] Solve the equation $\cos 7 x+\cos 3 x+\sin 7 x-\sin 3 x+\sqrt{2} \cos 4 x=0$.
Answer: $x=-\frac{\pi}{8}+\frac{\pi k}{2}, x=-\frac{\pi}{4}+\frac{2 \pi k}{3}, x=\frac{3 \pi}{28}+\frac{2 \pi k}{7}, k \in \mathbb{Z}$.

Solution. The given equation is equivalent to the following:

$2 \cos 5 x \cos 2 x+2 \sin 2 x \cos 5 x=-\sqrt{2} \cos 4 x \Leftrightarrow 2 \cos 5 x(\cos 2 x+\sin 2 x)=-\sqrt{2}\left(\cos ^{2} 2 x-\sin ^{2} 2 x\right) \Leftrightarrow$

![](https://cdn.mathpix.com/cropped/2024_05_06_ffd4340bf000b94108d2g-19.jpg?height=305&width=1701&top_left_y=1295&top_left_x=246)

Finally, we get $x=-\frac{\pi}{8}+\frac{\pi k}{2}, x=-\frac{\pi}{4}+\frac{2 \pi k}{3}, x=\frac{3 \pi}{28}+\frac{2 \pi k}{7}, k \in \mathbb{Z}$.