# How do you verify the identity costheta/(1+sintheta)+costheta/(1-sintheta)=2sectheta?

Aug 18, 2016

$L . H . S = R . H . S$

#### Explanation:

$L . H . S = \cos \frac{\theta}{1 + \sin \theta} + \cos \frac{\theta}{1 - \sin \theta}$

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

$= \frac{\cos \theta - \cos \theta \cdot \sin \theta + \cos \theta + \cos \theta \cdot \sin \theta}{\cos} ^ 2 \theta$

$= \frac{\cos \theta - \cancel{\cos \theta \cdot \sin \theta} + \cos \theta + \cancel{\cos \theta \cdot \sin \theta}}{\cos} ^ 2 \theta$

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

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

$= 2 \sec \theta$

$L . H . S = R . H . S$