Simplify the expression: $\dfrac{\sin 3 \alpha + \sin 5 \alpha + \sin 7 \alpha}{\cos 3 \alpha + \cos 5 \alpha + \cos 7 \alpha}$
$\dfrac{\sin 3 \alpha + \sin 5 \alpha + \sin 7 \alpha}{\cos 3 \alpha + \cos 5 \alpha + \cos 7 \alpha}$
$= \dfrac{\left(\sin 3 \alpha + \sin 7 \alpha\right) + \sin 5 \alpha}{\left(\cos 3 \alpha + \cos 7 \alpha\right) + \cos 5 \alpha}$
$= \dfrac{2 \sin \left(\dfrac{7 \alpha + 3 \alpha}{2}\right) \cos \left(\dfrac{7 \alpha - 3 \alpha}{2}\right) + \sin 5 \alpha}{2 \cos \left(\dfrac{7 \alpha + 3 \alpha}{2}\right) \cos \left(\dfrac{7 \alpha - 3 \alpha}{2}\right) + \cos 5 \alpha}$
$= \dfrac{2 \sin 5 \alpha \cos 2 \alpha + \sin 5 \alpha}{2 \cos 5 \alpha \cos 2 \alpha + \cos 5 \alpha}$
$= \dfrac{\sin 5 \alpha \left(2 \cos 2 \alpha + 1\right)}{\cos 5 \alpha \left(2 \cos 2 \alpha + 1\right)}$
$= \dfrac{\sin 5 \alpha}{\cos 5 \alpha}$
$= \tan 5 \alpha$