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Question
Prove the following identity :
`(1 + cotA + tanA)(sinA - cosA) = secA/(cosec^2A) - (cosecA)/sec^2A`
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Solution
LHS = `(1 + cotA + tanA)(sinA - cosA)`
= `(1 + cosA/sinA + sinA/cosA)(sinA - cosA)`
= `((sinAcosA + cos^2A + sin^2A)/(sinAcosA))(sinA - cosA)`
= `((sin^3A - cos^3A))/(sinAcosA)` (∵(`sin^3A - cos^3A) = (sinA - cosA)(sinA cosA + cos^2A + sin^2A`))
= `sin^3A/(sinAcosA) - cos^3A/(sinAcosA)`
= `sin^2A/cosA - cos^2A/sinA = 1/cosA xx sin^2A - 1/sinA xx cos^2A`
= `secAsin^2A - cosecAcos^2A`
= `secA/(cosec^2A) - (cosecA)/sec^2A`
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