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

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Answer 1 · 2015-08-21T20:33:29

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

$f (x) = 2 sin x cos x$ $f(x) = 2 sin x cos x$

For any $x$ $x$ in the domain of $f$ $f$ (the domain is $RR$ ), $-x$ is also in the domain of $f$ , and:

$f(-x) = 2sin(-x)cos(-x)$

$= 2(-sinx)(cosx)$

$= -2sinxcosx$

$= -f(x)$

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

Alternative

You could observe that $f(x) = 2sinxcosx = sin(2x)$ , so

$f(-x) = sin(2(-x)) = sin(-2x) = -sin(2x) = -f(x)$

Is the function 2 sin x cos x2sinxcosx2 sin x cos x even, odd or neither?