# How do you simplify cosx/(1+sinx) + (1+sinx)/cosx?

May 23, 2018

$\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x} = 2 \sec x$

#### Explanation:

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

$\textcolor{w h i t e}{\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}} = \frac{{\cos}^{2} x + {\sin}^{2} x + 1 + 2 \sin x}{\left(1 + \sin x\right) \cos x}$

$\textcolor{w h i t e}{\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}} = \frac{2 + 2 \sin x}{\left(1 + \sin x\right) \cos x}$

$\textcolor{w h i t e}{\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}} = \frac{2 \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(1 + \sin x\right)}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left(1 + \sin x\right)}}} \cos x}$

$\textcolor{w h i t e}{\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}} = \frac{2}{\cos} x$

$\textcolor{w h i t e}{\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}} = 2 \sec x$