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Question
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
Solution
`(cotA + cosecA - 1)/(cotA - cosecA + 1)`
= `(cotA + cosecA - (cosec^2A - cot^2A))/(cotA - cosecA + 1)` [`cosec^2A - cot^2A = 1`]
= `(cotA + cosecA - [(cosecA - cotA)(cosecA + cotA)])/(cotA - cosecA + 1)`
= `(cotA + cosecA[1 - cosecA + cotA])/(cotA - cosecA + 1)`
= `cotA + cosecA`
= `cosA/sinA + 1/sinA`
= `(1 + cosA)/sinA`
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