11.1. Solve the equation $\cos ^{2}(\sqrt{2} x)-\sin ^{2} x=1$.
Answer: $x=0$. Solution. The left side of the equation does not exceed one, and it can equal one only if the following two conditions are simultaneously satisfied: $\cos ^{2}(\sqrt{2} x)=1$ and $\sin x=0$. Hence, $\sqrt{2} x=\pi n$ and $x=\pi k, k, n \in \mathbf{Z}$. Then $\frac{\pi n}{\sqrt{2}}=\pi k$, i.e., $n=\sqrt{2} k$ for some integers $n, k$. Since $\sqrt{2}$ is an irrational number, it follows from the last equation that $k$ must be zero. Therefore, $k=n=0$, and $x=0$.