2. [5 points] Solve the equation $\cos 11 x - \cos 3 x - \sin 11 x + \sin 3 x = \sqrt{2} \cos 14 x$.
Answer: $x=-\frac{\pi}{28}+\frac{\pi k}{7}, x=\frac{\pi}{12}+\frac{2 \pi k}{3}, x=\frac{5 \pi}{44}+\frac{2 \pi k}{11}, k \in \mathbb{Z}$.

Solution. The given equation is equivalent to the following:

$-2 \sin 4 x \sin 7 x-2 \sin 4 x \cos 7 x=\sqrt{2} \cos 14 x \Leftrightarrow-2 \sin 4 x(\sin 7 x+\cos 7 x)=\sqrt{2}\left(\cos ^{2} 7 x-\sin ^{2} 7 x\right) \Leftrightarrow$

![](https://cdn.mathpix.com/cropped/2024_05_06_ffd4340bf000b94108d2g-01.jpg?height=283&width=1719&top_left_y=1315&top_left_x=226)

Finally, we get $x=-\frac{\pi}{28}+\frac{\pi k}{7}, x=\frac{\pi}{12}+\frac{2 \pi k}{3}, x=\frac{5 \pi}{44}+\frac{2 \pi k}{11}, k \in \mathbb{Z}$.