1. Desarrollo de la tangente de la suma:
$$ \tan \left( \frac{\pi}{4} + \frac{\alpha}{2} \right) = \frac{\tan \frac{\pi}{4} + \tan \frac{\alpha}{2}}{1 - \tan \frac{\pi}{4} \tan \frac{\alpha}{2}} = \frac{1 + \tan \frac{\alpha}{2}}{1 - \tan \frac{\alpha}{2}} $$

2. Expresión en senos y cosenos:
$$ \frac{1 + \frac{\sin(\alpha/2)}{\cos(\alpha/2)}}{1 - \frac{\sin(\alpha/2)}{\cos(\alpha/2)}} = \frac{\cos \frac{\alpha}{2} + \sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2} - \sin \frac{\alpha}{2}} $$

3. Transformación del segundo factor:
Sabemos que $1 - \sin \alpha = (\cos \frac{\alpha}{2} - \sin \frac{\alpha}{2})^2$ y $\cos \alpha = \cos^2 \frac{\alpha}{2} - \sin^2 \frac{\alpha}{2}$.
$$ \frac{1 - \sin \alpha}{\cos \alpha} = \frac{(\cos \frac{\alpha}{2} - \sin \frac{\alpha}{2})^2}{(\cos \frac{\alpha}{2} - \sin \frac{\alpha}{2})(\cos \frac{\alpha}{2} + \sin \frac{\alpha}{2})} = \frac{\cos \frac{\alpha}{2} - \sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2} + \sin \frac{\alpha}{2}} $$

4. Multiplicación de ambos términos:
$$ \left( \frac{\cos \frac{\alpha}{2} + \sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2} - \sin \frac{\alpha}{2}} \right) \cdot \left( \frac{\cos \frac{\alpha}{2} - \sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2} + \sin \frac{\alpha}{2}} \right) = 1 $$

Resultado:
$$ \boxed{1} $$