1. Find all real numbers $x$ and $y$ that satisfy the equation

$$
\log _{2}\left(\cos ^{2}(x y)+\frac{1}{\cos ^{2}(x y)}\right)=\frac{1}{y^{2}-2 y+2}
$$
Solution. For the expression on the left side to be well-defined, we need $x y \neq \frac{\pi}{2}+k \pi, k \in \mathbb{Z}$. Since $\cos ^{2}(x y)+\frac{1}{\cos ^{2}(x y)} \geq 2$, it follows that

$$
\log _{2}\left(\cos ^{2}(x y)+\frac{1}{\cos ^{2}(x y)}\right) \geq \log _{2} 2=1
$$

However, $\frac{1}{y^{2}-2 y+2}=\frac{1}{(y-1)^{2}+1} \leq 1$, so equality holds only if $\cos (x y)=\frac{1}{\cos (x y)}$ and $\frac{1}{(y-1)^{2}+1}=1$. From the second equation, we have $y=1$ and from the first equation, $x=k \pi, k \in \mathbb{Z}$. Therefore, the solutions are all pairs $(k \pi, 1)$, where $k \in \mathbb{Z}$.