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Question
Prove that:
`(cosecA - sinA)(secA - cosA) = 1/(tanA + cotA)`
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Solution
L.H.S. = `(cosecA - sinA)(secA - cosA)`
= `(1/sinA - sinA)(1/cosA - cosA)`
= `((1 - sin^2A)/sinA)((1 - cos^2A)/cosA)`
= `(cos^2A/sinA)(sin^2A/cosA)`
= sin A cos A
R.H.S. = `1/(tanA + cotA)`
= `1/(sinA/cosA + cosA/sinA)`
= `1/((sin^2A + cos^2A)/(sinAcosA))`
= `(sinAcosA)/(sin^2A + cos^2A)`
= `(sinAcosA)/1`
= sin A cos A
∴ L.H.S. = R.H.S.
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