2. Solve the equation $\left(\cos \frac{2 x}{5}-\cos \frac{2 \pi}{15}\right)^{2}+\left(\sin \frac{2 x}{3}-\sin \frac{4 \pi}{9}\right)^{2}=0$.
Solution. The equation is equivalent to the system of equations $\left\{\begin{array}{l}\cos \frac{2 x}{5}-\cos \frac{2 \pi}{15}=0 \\ \sin \frac{2 x}{3}-\sin \frac{4 \pi}{9}=0\end{array}\right.$.

Solve the equation $\cos \frac{2 x}{5}=\cos \frac{2 \pi}{15}:\left[\begin{array}{c}\frac{2 x}{5}=\frac{2 \pi}{15}+2 \pi k, \rightarrow x_{1}=\frac{\pi}{3}+5 \pi k, k \in Z \\ \frac{2 x}{5}=-\frac{2 \pi}{15}+2 \pi k \rightarrow x_{2}=-\frac{\pi}{3}+5 \pi n, n \in Z\end{array}\right.$.

Solve the equation $\sin \frac{2 x}{3}=\sin \frac{4 \pi}{9}:\left[\begin{array}{l}\frac{2 x}{3}=\frac{4 \pi}{9}+2 \pi m, \rightarrow x_{3}=\frac{2 \pi}{3}+3 \pi m, m \in Z \\ \frac{2 x}{3}=\frac{5 \pi}{9}+2 \pi s \rightarrow x_{4}=\frac{5 \pi}{6}+3 \pi s, s \in Z\end{array}\right.$.

The solutions to the original equation can be the intersections of the series $x_{1} \cap x_{3}, x_{1} \cap x_{4}, x_{2} \cap x_{3}, x_{2} \cap x_{4}$

Case 1. Intersection $x_{1} \cap x_{3}=\varnothing$. Indeed,

$$
x_{1}=\frac{\pi}{3}+5 \pi k=x_{3}=\frac{2 \pi}{3}+3 \pi m \rightarrow \frac{1}{3}=5 k-3 m .
$$

The equality is impossible, since the right side is an integer.

Case 2. Intersection $x_{1} \cap x_{4}=\varnothing$. Indeed,

$$
x_{1}=\frac{\pi}{3}+5 \pi k=x_{4}=\frac{5 \pi}{6}+3 \pi s \rightarrow \frac{1}{2}=5 k-3 s .
$$

The last equality is impossible, since $k$ and $s$ are integers.

Case 3. Intersection $x_{2} \cap x_{3} \neq \varnothing$. Indeed,

$$
x_{2}=-\frac{\pi}{3}+5 \pi n=x_{3}=\frac{2 \pi}{3}+3 \pi m \rightarrow 5 n-3 m=1 \rightarrow\left\{\begin{array}{c}
n=2+3 t \\
m=3+5 t, t \in Z
\end{array}\right.
$$

Then $x=-\frac{\pi}{3}+5 \pi(2+3 t)=\frac{29 \pi}{3}+15 \pi t, t \in Z$

Case 4. Intersection $x_{2} \cap x_{4}=\varnothing$. Indeed,

$$
x_{2}=-\frac{\pi}{3}+5 \pi n=x_{4}=\frac{5 \pi}{6}+3 \pi s \rightarrow 5 n-3 s=\frac{7}{6} \text {, which is impossible. }
$$

Answer. $x=\frac{29 \pi}{3}+15 \pi t, t \in Z$