# For what values of x does sqrt((1-sinx)/(1+sinx))=secx-tanx hold?

Apr 16, 2017

Identity holds for all values of $x$, irrespective of quadrant.

#### Explanation:

$\sqrt{\frac{1 - \sin x}{1 + \sin x}}$

= sqrt(((1-sinx)^2)/((1+sinx)(1-sinx))

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

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

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

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

= $\sec x - \tan x$

Above is an identity and hence it holds for all values of $x$ i.e. for any quadrant.