11.2. It is known that $\frac{1}{\cos (2022 x)}+\operatorname{tg}(2022 x)=\frac{1}{2022}$.

Find $\frac{1}{\cos (2022 x)}-\operatorname{tg}(2022 x)$.
Answer. 2022.

## Solution.

Consider the product $\left(\frac{1}{\cos \alpha}+\operatorname{tg} \alpha\right)\left(\frac{1}{\cos \alpha}-\operatorname{tg} \alpha\right)=\frac{1}{\cos ^{2} \alpha}-\operatorname{tg}^{2} \alpha=\frac{1-\sin ^{2} \alpha}{\cos ^{2} \alpha}=1$. Therefore, the desired expression is 2022.

Comment. A correct answer without justification - 0 points.