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Question
Prove the following identity :
`(tanθ + secθ - 1)/(tanθ - secθ + 1) = (1 + sinθ)/(cosθ)`
Solution
LHS = `(tanθ + secθ - 1)/(tanθ - secθ + 1) `
= `(tanθ + secθ - {sec^2θ - tan^2θ})/(1 + tanθ -secθ)`
= `(tanθ + secθ - {(secθ + tanθ)(secθ - tanθ)})/(1 + tanθ - secθ)`
= `([tanθ + secθ]{1 - (secθ - tanθ)})/[[1 + tanθ - secθ]` = `([tanθ + secθ][1 + tanθ - secθ])/[[1 + tanθ - secθ]]`
= `[tanθ + secθ] = (1 + sinθ)/cosθ` = RHS
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