Answer
$-2 \sin 4 \theta \sin \theta$
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 $(d)$, we have:
$\cos 5 \theta - \cos 3 \theta =- 2 \sin \dfrac{5 \theta + 3 \theta}{2} \ \sin \dfrac{5 \theta - 3 \theta }{2} \\=-2 \sin 4 \theta \sin \theta$
