Question

cos A - sin A +1 ÷ COS A + sin A - 1
= cosec A + cot A,
​

Answer 1

we have to prove : (cosA - sinA + 1)/(cosA + sinA - 1) = cosecA + cotA.

LHS = (cosA - sinA + 1)/(cosA + sinA - 1)

= (cosA - sinA + 1)(cosA - sinA + 1)/(cosA + sinA - 1)(cosA - sinA + 1)

= (cosA - sinA + 1)²/(cos²A - sin²A - 1 + 2sinA)

= (cosA - sinA + 1)²/(-2sin²A + 2sinA)

= (cos²A + sin²A - 2sinA.cosA + 2cosA - 2sinA + 1)/2sinA(1 - sinA)

= (2 - 2sinA.cosA + 2cosA - 2sinA)/2sinA(1 - sinA)

= (1 - sinA)(1 + cosA)/sinA(1 - sinA)

= (1 + cosA)/sinA

= cosecA + cotA = RHS

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Cos A- sin A+1/cos A+ sin A-1=cosec A+cot A

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