Question

Is the function `2 sin x cos x` even, odd or neither?

Answer 1

`f(x) = 2 sin x cos x` is an odd function.

Explanation:

`f(x) = 2 sin x cos x`

For any `x` in the domain of `f` (the domain is `RR`), `-x` is also in the domain of `f`, and:

`f(-x) = 2sin(-x)cos(-x)`

`= 2(-sinx)(cosx)`

`= -2sinxcosx`

`= -f(x)`

Since `f(-x) = -f(x)` for all `x` in the domain, `f` is odd.

Alternative

You could observe that `f(x) = 2sinxcosx = sin(2x)`, so

`f(-x) = sin(2(-x)) = sin(-2x) = -sin(2x) = -f(x)`