Example: Find the maximum value of the following expression $x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}$.
By Cauchy-Schwarz inequality and the mean inequality, we have
$$
\begin{array}{l}
\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)^{2} \leqslant\left(x^{2}+y^{2}\right)\left(1-y^{2}+1-x^{2}\right)=\left(x^{2}+y^{2}\right)\left(2-x^{2}-y^{2}\right) \\
\leqslant\left[\frac{\left(x^{2}+y^{2}\right)+\left(2-x^{2}-y^{2}\right)}{2}\right]^{2}=1 .
\end{array}
$$

Thus, the maximum value does not exceed 1. Furthermore, when $x \geqslant 0, y \geqslant 0$, and $x^{2}+y^{2}=1$, $x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}$ $=x^{2}+y^{2}=1$, hence the maximum value is 1.