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Question
Prove the following identity :
`(tanθ + sinθ)/(tanθ - sinθ) = (secθ + 1)/(secθ - 1)`
Solution
LHS = `(tanθ + sinθ)/(tanθ - sinθ)`
= `(sinθ/cosθ + sinθ)/(sinθ/cosθ - sinθ) = (sinθ + sinθcosθ)/(sinθ + sinθcosθ)`
= `(sinθ(1 + cosθ))/sin(1 + cosθ) = (1 + cosθ)/(1 - cosθ)`
= `(1 + 1/secθ)/(1 - 1/secθ) = ((secθ + 1)/secθ)/((secθ - 1)/secθ)`
= `(secθ + 1)/(secθ - 1)`
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