[tex]\large\underline{\sf{Solution-}}[/tex]
We have to evaluate the given expression.
[tex] \rm = \sqrt{ \dfrac{1 - \sin(x) }{1 + \sin(x) } } [/tex]
If we multiple both numerator and denominator by 1 - sin(x), then the value remains same. Let's do that.
[tex] \rm = \sqrt{ \dfrac{[1 - \sin(x)][1 - \sin(x) ]}{[1 + \sin(x)][1 - \sin(x) ]} } [/tex]
[tex] \rm = \sqrt{ \dfrac{[1 - \sin(x)]^{2}}{1- \sin^{2} (x) } } [/tex]
We know that:
[tex] \rm \longmapsto { \sin}^{2}(x) + \cos^{2}(x) = 1[/tex]
[tex] \rm \longmapsto \cos^{2}(x) = 1 - { \sin}^{2}(x)[/tex]
Therefore, the expression becomes:
[tex] \rm = \sqrt{ \dfrac{[1 - \sin(x)]^{2}}{\cos^{2} (x)}} [/tex]
[tex] \rm = \dfrac{1 - \sin(x)}{\cos(x)}[/tex]
[tex] \rm = \dfrac{1}{\cos(x)} - \dfrac{ \sin(x) }{ \cos(x) } [/tex]
[tex] \rm = \sec(x) - \tan(x) [/tex]
