The value of <span class="math-tex">\(\rm \tan \left(45 + \frac{\theta}{2} \right )\)</span> is
\[ \tan \left( 45^\circ + \frac{\theta}{2} \right) = \frac{1 + \tan \frac{\theta}{2}}{1 - \tan \frac{\theta}{2}} \] \[ = \frac{\sin \frac{\theta}{2} + \cos \frac{\theta}{2}}{\cos \frac{\theta}{2} - \sin \frac{\theta}{2}} = \frac{\left( \sin \frac{\theta}{2} + \cos \frac{\theta}{2} \right)^2}{\cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2}} \] \[ = \frac{1 + \sin \theta}{\cos \theta} = \sec \theta + \tan \theta \]
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