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