Always even (unless its domain is empty).

##color(white)()##Odd times odd

Suppose ##f(x)## and ##g(x)## are odd functions and ##h(x) = f(x)g(x)##

By definition:

##f(-x) = -f(x)## and ##g(-x) = -g(x)## for all ##x##

So:

##h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x)##

##= h(x)## for all ##x##

So ##h(x)## is even.

##color(white)()##Even times even

Now suppose that ##f(x)## and ##g(x)## are even functions and ##h(x) = f(x)g(x)##

By definition:

##f(-x) = f(x)## and ##g(-x) = g(x)## for all ##x##

So:

##h(-x) = f(-x)g(-x) = f(x)g(x) = h(x)## for all ##x##

So ##h(x)## is even.

##color(white)()##Exception

If the domains of ##f(x)## and ##g(x)## do not intersect then their product ##f(x)g(x)## has an empty domain so is the empty function. The empty function probably does not count as odd or even.

For example:

Let ##f(x) = cos^(-1)(x)## and ##g(x) = sec^(-1)(x/2)##.

Then the domain of ##f(x)## is ##[-1 1]## and the domain of ##g(x)## is ##(-oo -2] uu [2 oo)##. They are both even functions.

The domain of ##f(x)g(x)## is ##[-1 1] nn ((-oo 2] uu [2 oo)) = O/##