# Question #ccc87

Sep 22, 2016

$\cos \frac{\theta}{1 + \sin \theta} + \frac{1 + \sin \theta}{\cos} \theta$

$= \frac{\cos \theta \left(1 - \sin \theta\right)}{\left(1 - \sin \theta\right) \left(1 + \sin \theta\right)} + \frac{1 + \sin \theta}{\cos} \theta$

$= \frac{\cos \theta \left(1 - \sin \theta\right)}{1 - {\sin}^{2} \theta} + \frac{1 + \sin \theta}{\cos} \theta$

$= \frac{\cos \theta \left(1 - \sin \theta\right)}{\cos} ^ 2 \theta + \frac{1 + \sin \theta}{\cos} \theta$

$= \frac{1 - \sin \theta}{\cos} \theta + \frac{1 + \sin \theta}{\cos} \theta$

$= \frac{1}{\cos} \theta - \sin \frac{\theta}{\cos} \theta + \frac{1}{\cos} \theta + \sin \frac{\theta}{\cos} \theta$

$= \frac{2}{\cos} \theta = 2 \sec \theta$

Alternative

$\cos \frac{\theta}{1 + \sin \theta} + \frac{1 + \sin \theta}{\cos} \theta$

$= \frac{{\cos}^{2} \theta + {\left(1 + \sin \theta\right)}^{2}}{\left(1 + \sin \theta\right) \cos \theta}$

$= \frac{{\cos}^{2} \theta + \left(1 + 2 \sin \theta + {\sin}^{2} \theta\right)}{\left(1 + \sin \theta\right) \cos \theta}$

$= \frac{2 + 2 \sin \theta}{\left(1 + \sin \theta\right) \cos \theta}$

$= \frac{2}{\cos} \theta = 2 \sec \theta$