Step-by-step Solution:

\[ \tan \left( 45^\circ + \frac{\theta}{2} \right) = \frac{1 + \tan \frac{\theta}{2}}{1 - \tan \frac{\theta}{2}} \] \[ = \frac{\sin \frac{\theta}{2} + \cos \frac{\theta}{2}}{\cos \frac{\theta}{2} - \sin \frac{\theta}{2}} = \frac{\left( \sin \frac{\theta}{2} + \cos \frac{\theta}{2} \right)^2}{\cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2}} \] \[ = \frac{1 + \sin \theta}{\cos \theta} = \sec \theta + \tan \theta \]