Let $x$ and $y$ be real numbers.  Find the set of possible values of
\[\frac{(x + y)(1 - xy)}{(1 + x^2)(1 + y^2)}.\]
Question:
Let $x$ and $y$ be real numbers.  Find the set of possible values of
\[\frac{(x + y)(1 - xy)}{(1 + x^2)(1 + y^2)}.\]
Answer:
Let's think step by step.

user
\left[ -\frac{1}{2}, \frac{1}{2} \right]
Let $x$ and $y$ be real numbers.  Find the set of possible values of
\[\frac{(x + y)(1 - xy)}{(1 + x^2)(1 + y^2)}.\]
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