Find the largest value of the expression

$$
x y + x \sqrt{1-y^{2}} + y \sqrt{1-x^{2}} - \sqrt{\left(1-x^{2}\right)\left(1-y^{2}\right)}
$$
The expression is well-defined only for $|x|,|y| \leq 1$ and we can assume that $x, y \geq 0$. Let $x=\cos \alpha$ and $y=\cos \beta$ for some $0 \leq \alpha, \beta \leq \frac{\pi}{2}$. This reduces the expression to

$$
\cos \alpha \cos \beta+\cos \alpha \sin \beta+\cos \beta \sin \alpha-\sin \alpha \sin \beta=\cos (\alpha+\beta)+\sin (\alpha+\beta)=\sqrt{2} \cdot \sin \left(\alpha+\beta+\frac{\pi}{4}\right)
$$

which does not exceed $\sqrt{2}$. The equality holds when $\alpha+\beta+\frac{\pi}{4}=\frac{\pi}{2}$, for example when $\alpha=\frac{\pi}{4}$ and $\beta=0$, i.e., $x=\frac{\sqrt{2}}{2}$ and $y=1$.