Answer
$\cos \theta$
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 $(a)$, we have:
$\dfrac{\sin 3 \theta + \sin \theta }{2 \sin 2 \theta}=\dfrac{2 \sin \dfrac{ 3 \theta + \theta}{2} \ \cos \dfrac{3 \theta - \theta }{2} }{2 \sin 2 \theta}\\=\dfrac{2 \sin 2 \theta \ \cos \theta}{2 \sin 2 \theta} \\=\cos \theta$
