Example 2 Find the maximum value of
$$x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}$$
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 arithmetic mean inequality, we have
$$\left|x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right| \leqslant \frac{x^{2}+y^{2}+2-x^{2}-y^{2}}{2}=1 .$$

If $x=\frac{1}{2}, y=\frac{\sqrt{3}}{2}$, then
$$x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}=1$$

Thus, the maximum value sought is 1.