The problem asks us to prove that the product of two even functions is even, and the product of two odd functions is even.

1. Problem Description

The problem asks us to prove that the product of two even functions is even, and the product of two odd functions is even.

2. Solution Steps

Let $f(x)$ and $g(x)$ be two functions.

A function is even if $f(-x) = f(x)$.

A function is odd if $f(-x) = -f(x)$.

First, we prove that the product of two even functions is even.

Let $f(x)$ and $g(x)$ be two even functions. Then $f(-x) = f(x)$ and $g(-x) = g(x)$.

Let $h(x) = f(x)g(x)$. Then $h(-x) = f(-x)g(-x) = f(x)g(x) = h(x)$.

Since $h(-x) = h(x)$, the product of two even functions is even.

Second, we prove that the product of two odd functions is even.

Let $f(x)$ and $g(x)$ be two odd functions. Then $f(-x) = -f(x)$ and $g(-x) = -g(x)$.

Let $h(x) = f(x)g(x)$. Then $h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x)$.

Since $h(-x) = h(x)$, the product of two odd functions is even.

3. Final Answer

The product of two even functions is even, and the product of two odd functions is even.