Odd Function Times Odd Function is Even

Theorem
Let $S \subset \R$ be a symmetric set of real numbers:
 * $\forall x \in S: -x \in X$

Let $f, g: X \to \R$ be two odd functions.

Let $f \cdot g$ denote the pointwise product of $f$ and $g$.

Then $\paren {f \cdot g}: S \to \R$ is an even function.

Proof
Thus, by definition, $\paren {f \cdot g}$ is an even function.

Also see

 * Even Function Times Even Function is Even
 * Odd Function Times Even Function is Odd