Question

Prove the following trigonometric identities.

`sqrt((1 + sin A)/(1 - sin A)) = sec A + tan A`

Answer 1

We need to prove `sqrt((1 + sin A)/(1 - sin A)) = sec A + tan A`

Here, rationalising the L.H.S, we get

`sqrt((1 + sin A)/(1 - sin A)) = sqrt((1+ sin A)/(1 - sin A)) xx sqrt((1 + sin A)/(1 + sin A))`

`= sqrt((1 + sin A)^2/(1 - sin^2 A)`

Further using the property, `sin^2 theta + cos^2 theta = 1` we get

So,

`sqrt((1 - cos A)/(1 - cos^2 A)) = sqrt((1 -  cos A)^2/sin^2 A)`

`= (1 - cos A)/sin A`

`= 1/sin A - cos A/sin A`

`=cosec A  - cot A`

Hence proved.