Question

Prove that: `sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`

Prove the following trigonometric identities:

`sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`

Answer 1

LHS = `sqrt((1/cos theta - 1)/(1/cos theta + 1)) + sqrt((1/cos theta +1)/(1/cos theta - 1))`

`= sqrt(((1 - cos theta)/cos theta)/((1+ cos theta)/cos theta)) + sqrt(((1 + cos theta)/cos theta)/((1 - cos theta)/cos theta)`

`= sqrt((1 - cos theta)/(1 + cos theta)) +sqrt((1 + cos theta)/(1 - cos theta))`

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

`= sqrt((1 - cos theta)^2/(1 - cos^2 theta)) + sqrt((1 + cos theta)^2/(1 - cos^2 theta))`

`=(1 - cos theta)/sin theta + (1 + cos theta)/sin theta`

`= (1 - cos theta + 1 + cos theta)/sin theta`

`= 2/sin theta`

= 2 cosec