Question

Prove the following trigonometric identities.

`sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`

Answer 1

We need to prove  `sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`

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

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

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

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

So,

`sqrt((1 - cos A)^2/(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.