# How do you simplify  [(1 + sin theta)/cos theta] + [cos theta/(1 + sin theta)]?

Apr 25, 2016

$2 \sec \theta$

#### Explanation:

Multiply the fractions to achieve a common denominator.

$= \left[\frac{1 + \sin \theta}{\cos} \theta\right] \left[\frac{1 + \sin \theta}{1 + \sin \theta}\right] + \left[\cos \frac{\theta}{1 + \sin \theta}\right] \left[\cos \frac{\theta}{\cos} \theta\right]$

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

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

Recall the Pythagorean Identity ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$.

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

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

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

$= 2 \sec \theta$

Apr 25, 2016

$2 \sec \theta$

#### Explanation:

Begin by writing the fractions as a single fraction by extracting the lowest common denominator
In this case $\cos \theta \left(1 + \sin \theta\right)$

rArr( (1 +sintheta)^2 + cos^2 theta)/(costheta(1+sintheta)

Expanding the numerator

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

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

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

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

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