# How to solve ecuation: (sqrt((1-sinx)/(1+sinx))-sqrt((1+sinx)/(1-sinx)))(sqrt((1-cosx)/(1+cosx)-sqrt((1+cosx)/(1-cosx)=-4 ?

Jan 8, 2018

$\left(\sqrt{\frac{1 - \sin x}{1 + \sin x}} - \sqrt{\frac{1 + \sin x}{1 - \sin x}}\right) \left(\sqrt{\frac{1 - \cos x}{1 + \cos x}} - \sqrt{\frac{1 + \cos x}{1 - \cos x}}\right) = 4$ and not $- 4$

#### Explanation:

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

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

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

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

= $\left(- 2 \tan x\right) \times \left(- 2 \cot x\right)$

= $4$