Answer
$\dfrac{1}{2} [\sin (3 \theta) - \sin (2 \theta) ]$
Work Step by Step
Recall the Product to Sum Identities:
$a) \sin x \cos y =\dfrac{1}{2} [\sin (x-y) +\sin (x+y)] \\ b) \sin x \sin y =\dfrac{1}{2} [\cos (x-y) -\cos (x+y)] \\ c) \cos x \cos y =\dfrac{1}{2} [\cos (x-y) +\cos (x+y)]$
We know that $\sin$, $\csc$ and $\tan$ are odd trigonometric functions. This implies that $f (-x)=f(x) \implies \sin(-x)=- \sin(x)$
By identity $(a)$, we have:
$\sin ( \dfrac{\theta}{2}) \ \cos (\dfrac{5 \theta}{2})=\dfrac{1}{2} [\sin [( \dfrac{\theta}{2}) + ( \dfrac{5 \theta}{2}) ]+\sin[( \dfrac{\theta}{2}) - ( \dfrac{5 \theta}{2}) ] ] \\=\dfrac{1}{2} [\sin (3 \ \theta) + \sin (-2 \theta) ]\\=\dfrac{1}{2} [\sin (3 \theta) - \sin (2 \theta) ]$
