Question

How do you know if `cosx sinx` is an even or odd function?

Answer 1

Find `cos(-x) sin(-x) = - (cos(x) sin(x))`, so it's an odd function.

Explanation:

An even function is a function satisfying:

`f(-x) = f(x)` for any `x`

An odd function is a function satisfying:

`f(-x) = -f(x)` for any `x`

Note that `cos(x)` is even since `cos(-x) = cos(x)` for any `x`

Note that `sin(x)` is odd since `sin(-x) = -sin(x)` for any `x`

Let `f(x) = cos(x) sin(x)`

Then for any `x`:

`f(-x)`

`= cos(-x) * sin(-x)`

`= cos(x) * (-sin(x))`

`= - cos(x)sin(x)`

`= -f(x)`

So `f(x) = cos(x) sin(x)` is an odd function.

In general, if `e(x)` is an even function and `o(x)` is an odd function, then `e(x)o(x)` is odd too. The odd/even result you get when multiplying functions is like the result when you add even and odd numbers.

The functions `e(o(x))`, `o(e(x))` and `e(e(x))` will all be even, but the function `o(o(x))` will be odd. The odd/even result you get when composing functions is like the result when you multiply even and odd numbers.