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Question
Prove that:
`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`
Solution
`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = sqrt(sectheta - 1)/sqrt(sectheta + 1) + sqrt(sectheta + 1)/sqrt(sectheta - 1)`
= `(sqrt(sectheta - 1)sqrt(sectheta - 1) + sqrt(sectheta + 1)sqrt(sectheta + 1))/(sqrt(sectheta + 1)sqrt(sectheta - 1))`
= `((sqrt(sectheta - 1))^2 + (sqrt(sectheta + 1))^2)/sqrt((sectheta - 1)(sectheta + 1))`
= `(sectheta - 1 + sectheta + 1)/sqrt(sec^2theta - 1)`
= `(2sectheta)/sqrt(tan^2theta)`
= `(2sectheta)/tantheta`
= `(2 1/costheta)/(sintheta/costheta)`
= `2 1/sintheta`
= `2 cosectheta`
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