<s>[INST] Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$ I would appreciate if somebody could help me with the following problem
Let $|x|\leq1, |y|\leq 1$. Show that
$$
0  \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy  \sqrt{1-x^{2}}  \sqrt{1-y^{2}} \leq 1
$$
 [/INST] We need only note that
\begin{align*}
x^{2}+y^{2} -2x^{2} y^{2}+2xy  \sqrt{1-x^{2}}  \sqrt{1-y^{2}} &= \left(x\sqrt{1-y^2}+y\sqrt{1-x^2}\,\right)^2\\
&\le \left(\sqrt{(x^2+y^2)(2-x^2-y^2)}\,\right)^2\\
&\le \left(\frac{1}{2}(x^2+y^2 + 2-x^2-y^2)\,\right)^2\\
&=1.
\end{align*}
</s>