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Question
Prove the following identity :
`sqrt((secq - 1)/(secq + 1)) + sqrt((secq + 1)/(secq - 1))` = 2 cosesq
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Solution
`sqrt((secq - 1)/(secq + 1)) + sqrt((secq + 1)/(secq - 1))`
= `sqrt((secq - 1)/(secq + 1) . (secq - 1)/(secq - 1)) + sqrt((secq + 1)/(secq - 1) . (secq + 1)/(secq + 1))`
= `sqrt((secq - 1)^2/(sec^2q - 1)) + sqrt((secq + 1)^2/(secq^2 - 1)`
= `sqrt((secq - 1)^2/tan^2q) + sqrt((secq + 1)^2/(tan^2q)` (`Q sec^2q - 1 = tan^2q`)
= `(secq - 1)/tanq + (secq + 1)/tanq = (secq - 1 + secq + 1)/tanq`
= `(2secq)/tanq = (2/cosq)/(sinq/cosq) = 2/sinq = 2cosecq`
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