19. Find the maximum value of the expression $x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}$. (1990 Moscow Mathematical Olympiad problem)
19. By Cauchy-Schwarz inequality, we have
$$\left|x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right|^{2} \leqslant\left(x^{2}+y^{2}\right)\left(2-x^{2}-y^{2}\right)$$

By the AM-GM inequality, we have
$$\left(x^{2}+y^{2}\right)\left(2-x^{2}-y^{2}\right) \leqslant\left[\frac{\left(x^{2}+y^{2}\right)+\left(2-x^{2}-y^{2}\right)}{2}\right]^{2}=1$$

Taking \(x=\frac{1}{2}, y=\frac{\sqrt{3}}{2}\), we get \(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}=1\), so the maximum value is 1.