Simplify the expression: $\tan \left(\dfrac{\alpha}{2} + \dfrac{\pi}{4}\right) \left(\dfrac{1 - \sin \alpha}{\cos \alpha}\right)$
$\tan \left(\dfrac{\alpha}{2} + \dfrac{\pi}{4}\right) \left(\dfrac{1 - \sin \alpha}{\cos \alpha}\right)$
$= \left(\dfrac{\tan \dfrac{\pi}{4} + \tan \dfrac{\alpha}{2}}{1 - \tan \dfrac{\pi}{4} \times \tan \dfrac{\alpha}{2}}\right) \left(\dfrac{\sin^2 \dfrac{\alpha}{2} + \cos^2 \dfrac{\alpha}{2} - 2 \sin \dfrac{\alpha}{2} \cos \dfrac{\alpha}{2}}{\cos^2 \dfrac{\alpha}{2} - \sin^2 \dfrac{\alpha}{2}}\right)$
$= \left(\dfrac{1 + \tan \dfrac{\alpha}{2}}{1 - \tan \dfrac{\alpha}{2}}\right) \times \dfrac{\left(\cos \dfrac{\alpha}{2} - \sin \dfrac{\alpha}{2}\right)^2}{\left(\cos \dfrac{\alpha}{2} + \sin \dfrac{\alpha}{2}\right) \left(\cos \dfrac{\alpha}{2} - \sin \dfrac{\alpha}{2}\right)}$
$= \left(\dfrac{1 + \dfrac{\sin \dfrac{\alpha}{2}}{\cos \dfrac{\alpha}{2}}}{1 - \dfrac{\sin \dfrac{\alpha}{2}}{\cos \dfrac{\alpha}{2}}}\right) \times \left(\dfrac{\cos \dfrac{\alpha}{2} - \sin \dfrac{\alpha}{2}}{\cos \dfrac{\alpha}{2} + \sin \dfrac{\alpha}{2}}\right)$
$= \left(\dfrac{\cos \dfrac{\alpha}{2} + \sin \dfrac{\alpha}{2}}{\cos \dfrac{\alpha}{2} - \sin \dfrac{\alpha}{2}}\right) \times \left(\dfrac{\cos \dfrac{\alpha}{2} - \sin \dfrac{\alpha}{2}}{\cos \dfrac{\alpha}{2} + \sin \dfrac{\alpha}{2}}\right)$
$= 1$