# How do you provetanx/(1+cosx) + sinx/(1-cosx)=cotx+(secx)(cscx)?

Jul 16, 2016

To prove $\tan \frac{x}{1 + \cos x} + \sin \frac{x}{1 - \cos x} = \cot x + \left(\sec x\right) \left(\csc x\right)$

$L H S = \frac{\sin x \left(1 - \cos x\right)}{\cos x \left(1 + \cos x\right) \left(1 - \cos x\right)} + \frac{\sin x \left(1 + \cos x\right)}{\left(1 - \cos x\right) \left(1 + \cos x\right)}$

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

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

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

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

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

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

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

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

$= \cot x + \csc x \sec x = R H S$

Proved