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