2. For each value of the parameter $k$, solve the equation

$$
2+\cos ^{2} x+\cos 4 x+\cos 2 x+2 \sin 3 x \cdot \sin 7 x+\sin ^{2} 7 x=\cos ^{2}\left(\frac{\pi k}{2022}\right)
$$
Answer: $P r i=2022 \cdot m, m \in Z \quad x=\frac{\pi}{4}+\frac{\pi n}{2}, n \in Z$;

for other values of the parameter $k$ there are no solutions.

## Solution.

The original equation is equivalent to the following equation:

$$
\begin{gathered}
(\sin 3 x+\sin 7 x)^{2}+(\cos 3 x+\cos x)^{2}+\sin ^{2}\left(\frac{\pi k}{2022}\right)=0 \\
\left\{\begin{array}{l}
\sin 3 x+\sin 7 x=0 \\
\cos 3 x+\cos x=0 \\
\sin \left(\frac{\pi k}{2022}\right)=0
\end{array}\right.
\end{gathered}
$$

If $k=2022 \cdot m, m \in Z$, then solving the first and second equations of the system, we get

$$
x=\frac{\pi}{4}+\frac{\pi n}{2}, n \in Z
$$

For other values of the parameter $k$ there are no solutions.