# How do you verify the identity sin4x= 4sinxcosx(1-sin^2x)?

Apr 23, 2015

Remember
$\sin \left(2 \theta\right) = 2 \cdot \sin \left(\theta\right) \cdot \cos \left(\theta\right)$
and
$\cos \left(2 \theta\right) = 1 - 2 \cdot {\sin}^{2} \left(\theta\right)$

So
$\sin \left(4 x\right)$

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

$= 2 \cdot \left(2 \cdot \sin \left(x\right) \cdot \cos \left(x\right)\right) \cdot \left(1 - 2 \cdot {\sin}^{2} \left(x\right)\right)$

$= 4 \sin \left(x\right) \cos \left(x\right) \left(1 - {\sin}^{2} \left(x\right)\right)$