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

Oct 16, 2016

You have to use the definitions of even and odd functions

#### Explanation:

A function $f$ is even if it satisfies $f \left(- x\right) = f \left(x\right)$. For example $\cos \left(- x\right) = \cos x$, so $\cos x$ is even

A function $f$ is odd if it satisfies $f \left(- x\right) = - f \left(x\right)$. For example $\sin \left(- x\right) = - \sin x$, so $\sin x$ is odd

Lets see the given function $f \left(x\right) = 2 \sin x \cos x$, and evaluate $f \left(- x\right)$

$f \left(- x\right) = 2 \sin \left(- x\right) \cos \left(- x\right) = 2 \cdot \left(- \sin x\right) \cdot \cos \left(x\right)$, using the equalities above.

Then, $f \left(- x\right) = - 2 \sin x \cos x = - f \left(x\right)$, and so the function given is odd