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

Sep 23, 2015

$f \left(x\right) = \cos \left(x\right) \cdot \sin \left(x\right)$ is an odd function.

#### Explanation:

Recall that the definition of an even function is
$f \left(x\right) = f \left(- x\right)$
and the definition of an odd function is
$f \left(x\right) = - f \left(x\right)$

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

$f \left(- x\right) = \cos \left(- x\right) \cdot \sin \left(- x\right) =$
$= \cos \left(x\right) \cdot \left[- \sin \left(x\right)\right] = - \cos \left(x\right) \cdot \sin \left(x\right) = - f \left(x\right)$

Therefore, $f \left(x\right) = \cos \left(x\right) \cdot \sin \left(x\right)$ is an odd function.