11.1. Solve the system $\left\{\begin{array}{c}x^{2}+4 \sin ^{2} y-4=0 \\ \cos x-2 \cos ^{2} y-1=0\end{array}\right.$.
Answer: $x=0, y=\pi / 2+\pi k$, where $k-$ is an integer.

From the second equation, we get $2 \cos ^{2} y=\cos x-1 \leq 0$, from which it follows that $\cos ^{2} y=0, \sin ^{2} y=1$. Now from the first equation, we get $x=0$.

Remark. An incorrect answer or a correct answer without justification -0 points.