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

Aug 21, 2015

$f \left(x\right) = 2 \sin x \cos x$ is an odd function.

#### Explanation:

$f \left(x\right) = 2 \sin x \cos x$

For any $x$ in the domain of $f$ (the domain is $\mathbb{R}$), $- x$ is also in the domain of $f$, and:

$f \left(- x\right) = 2 \sin \left(- x\right) \cos \left(- x\right)$

$= 2 \left(- \sin x\right) \left(\cos x\right)$

$= - 2 \sin x \cos x$

$= - f \left(x\right)$

Since $f \left(- x\right) = - f \left(x\right)$ for all $x$ in the domain, $f$ is odd.

Alternative

You could observe that $f \left(x\right) = 2 \sin x \cos x = \sin \left(2 x\right)$, so

$f \left(- x\right) = \sin \left(2 \left(- x\right)\right) = \sin \left(- 2 x\right) = - \sin \left(2 x\right) = - f \left(x\right)$