Prove that: `sqrt((1-sintheta)/(1+sintheta))= sec theta- tan theta `when `(-pi/2) < theta < pi/2` and `-sec theta+tan theta` when `pi/2< theta< (3pi)/2`

Explanation

` sqrt{frac{1-sin theta}{1+sin theta}} `

`= sqrt{frac{1-sin theta}{1+sin theta} times frac{1-sin theta}{1-sin theta}} `

`= sqrt{frac{(1-sin theta)^{2}}{1-sin ^{2} theta}} `

`= sqrt{frac{(1-sin theta)^{2}}{cos ^{2} theta}}(cos ^{2} theta+sin ^{2} theta=1) `

`= frac{1-sin theta}{cos theta} `

`= frac{1}{cos theta}-frac{sin theta}{cos theta} `

`= sec theta-tan theta`