2. (2 points) Find the minimum value of the expression $x^{2}+4 x \sin y-4 \cos ^{2} y$.
Answer: -4.

Solution: Add and subtract $4 \sin ^{2} y$. We get the expression $x^{2}+4 x \sin y+4 \sin ^{2} y-4=(x+2 \sin y)^{2}-4$. It is clear that this expression has a minimum value of -4.

Idea of Solution 2: To find the minimum value, we can take the derivatives of this expression with respect to $x$ and $y$ and find the point where both these derivatives are zero. The derivative with respect to $x$ is zero when $x=-2 \sin y$.

The derivative with respect to $y$ is $4 x \cos y+8 \sin y \cos y$. It is zero in two cases: when $x=-2 \sin y$ and when $\cos y=0$. After this, it is necessary to determine which of these points are indeed local minima and why a global minimum exists.

A rigorous proof of this in terms of derivatives unfortunately goes beyond the school curriculum, so for school students, this solution can serve more as a hint to the answer and to the solution presented above.