Question

How do you determine if `2 sin x cos x` is an even or odd function?

Answer 1

You have to use the definitions of even and odd functions

Explanation:

A function `f` is even if it satisfies `f(-x)=f(x)`. For example `cos(-x)=cosx`, so `cosx` is even

A function `f` is odd if it satisfies `f(-x)=-f(x)`. For example `sin(-x)=-sinx`, so `sinx` is odd

Lets see the given function `f(x)=2 sinx cosx`, and evaluate `f(-x)`

`f(-x)= 2 sin(-x) cos(-x)=2*(-sinx) * cos(x)`, using the equalities above.

Then, `f(-x)=- 2sinx cosx=-f(x)`, and so the function given is odd