Answer
$\tan 3 \ \theta$
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 identities $(a)$ and (c), we have:
$\dfrac{\sin 4 \theta + \sin 2 \theta }{\cos 3 \theta + \cos \theta}=\dfrac{2 \sin \dfrac{ 4 \theta + 2 \theta}{2} \ \cos \dfrac{4 \theta - 2\theta }{2} }{2 \cos \dfrac{ 3 \theta + \theta}{2} \ \cos \dfrac{3 \theta - \theta }{2} }\\=\dfrac{2 \sin 3 \theta \ \cos \theta }{2 \cos 3 \theta \ \cos \theta } \\=\tan 3 \ \theta$