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Question
Prove the following identities.
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
Solution
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
`sqrt((1 + sin theta)/(1 - sin theta)) = sqrt(((1 + sin theta)(1 + sin theta))/((1 - sin theta)(1 + sin theta))`
= `sqrt((1 + sin theta)^2/(1 - sin^2 theta)`
= `sqrt((1 + sin theta)^2/(cos^2 theta)`
= `(1 + sin theta)/cos theta`
`sqrt(((1 - sin theta))/((1 + sin theta))) = sqrt(((1 - sin theta))/((1 - sin theta)) xx ((1 + sin theta))/((1 - sin theta))`
= `sqrt((1 - sin theta)^2/(1 - sin^2 theta)`
= `sqrt((1- sin theta)^2/(cos^2 theta)) = (1 - sin theta)/cos theta`
L.H.S. = `sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta)`
= `(1 + sin theta)/cos theta + (1 - sin theta)/cos theta`
= `(1 + sin theta + 1 - sin theta)/cos theta`
= `2/cos theta`
= 2 sec θ
L.H.S. = R.H.S.
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