Question 107
Prove that
$$
-\frac{1}{2} \leq \frac{(x+y)(1-x y)}{\left(1+x^{2}\right)\left(1+y^{2}\right)} \leq \frac{1}{2}
$$
for all real numbers $x$ and $y$.
$$
-\frac{1}{2} \leq \frac{(x+y)(1-x y)}{\left(1+x^{2}\right)\left(1+y^{2}\right)} \leq \frac{1}{2}
$$
for all real numbers $x$ and $y$.