# How do you prove (cosx/(1+sinx))+((1+sinx)/cosx)=2secx?

Jul 23, 2015

Convert the left side into terms with common denominator and add (converting ${\cos}^{2} + {\sin}^{2}$ to $1$ along the way); simplify and refer to definition of $\sec = \frac{1}{\cos}$

#### Explanation:

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

= (cos^2(x) + 1+2sin(x) + sin^2(x))/(cos(x)( 1+sin(x)

= (2 +2sin(x))/(cos(x)(1+sin(x))

$= \frac{2}{\cos} \left(x\right)$

$= 2 \cdot \frac{1}{\cos} \left(x\right)$

$= 2 \sec \left(x\right)$