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

Apr 15, 2015

$L H S$

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

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

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

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

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

$= \frac{2}{\sin} x$

$= 2 \csc x$

$= R H S$

This is because:

$\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$

And also because:

${\cos}^{2} x + {\sin}^{2} x = 1$