Answer
$2 \sin \theta \cos 3 \theta$
Work Step by Step
Recall the Sum to Product Identities:
$a) \sin x +\sin y =2 \sin \dfrac{x+y}{2} \cos \dfrac{x-y}{2} \\ b) \sin x - \sin y =2 \sin \dfrac{x -y}{2} \cos \dfrac{x +y}{2} \\ c) \cos x +\cos y =2 \cos \dfrac{x+y}{2} \cos \dfrac{x-y}{2} \\ d) \cos x +\cos y = -2 \sin \dfrac{x+y}{2} \sin \dfrac{x-y}{2}$
By identity $(b)$, we have:
$\sin 4 \theta -\sin 2 \theta =2 \sin \dfrac{4 \theta - 2 \theta}{2} \ \cos \dfrac{4 \theta + 2 \theta }{2} \\=2 \sin \theta \cos 3 \theta$
