Question

Is the function `f(x) =cosx sinx` even, odd or neither?

Answer 1

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

Explanation:

Recall that the definition of an even function is
`f(x) = f(-x)`
and the definition of an odd function is
`f(x) = -f(x)`

Let's check either of these properties for our function
`f(x) = cos(x)*sin(x)`
taking into account that `cos(x)` is an even function because
`cos(x) = cos(-x)`
and `sin(x)` is an odd function because
`sin(-x) = -sin(x)`

`f(-x) = cos(-x) * sin(-x) =`
`= cos(x) * [-sin(x)] = -cos(x) * sin(x) = -f(x)`

Therefore, `f(x) = cos(x)*sin(x)` is an odd function.