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Question
Prove the following identities:
`(sinA - cosA + 1)/(sinA + cosA - 1) = cosA/(1 - sinA)`
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Solution
`(sinA-cosA+1)/(sinA+cosA-1)`
= `(sinA - cosA + 1)/(sinA + cosA - 1) xx (sinA - (cosA - 1))/(sinA - (cosA - 1))`
= `(sinA - cosA + 1)^2/(sin^2A - (cosA - 1)^2)`
= `(sin^2A + cos^2A + 1 - 2sinAcosA - 2cosA + 2sinA)/(sin^2A - cos^2A - 1 + 2cosA)`
= `(1 + 1 - 2sinAcosA - 2cosA + 2sinA)/(-cos^2A - cos^2A + 2cosA)`
= `(2(1 - cosA) + 2sinA(1 - cosA))/(2cosA(1 - cosA)`
= `(1 + sinA)/cosA`
= `(1 + sinA)/cosA xx (1 - sinA)/(1 - sinA)`
= `cos^2A/(cosA(1 - sinA))`
= `cosA/(1 - sinA)`
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