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Question
Prove the following identity :
`(cosecA - sinA)(secA - cosA) = 1/(tanA + cotA)`
Solution
LHS = `(cosecA - sinA)(secA - cosA)`
= `(1/sinA - sinA)(1/cosA - cosA)`
= `((1-sin^2A)/(sinA))((1 - cos^2A)/cosA)`
= `(cos^2A/sinA)(sin^2A/cosA)` = cosA.sinA
RHS = `1/(tanA + cotA)`
= `1/(sinA/cosA + cosA/sinA) = 1/((sin^2A + cos^2A)/(sinA.cosA))` = cosA.sinA
Hence , LHS = RHS
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