Question

prove that cot A+cosecA-1 / cot A-cosec A+1 = 1 +cos A/sinA. use cos A/sinA =cot A​

Answer 1

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Answer 2

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Step-by-step explanation:

To prove that (cot A + cosec A -1) / (cot A - cosec A +1) = (1+cos A) / sin A

LHS=(cos A / sin A + 1/sin A - sin A / sin A) / (cos A / sin A - 1 / sin A + sin A / sin A)

= (cos A + 1 - sin A) / (cos A - 1 + sin A) * (cos A + 1 - sin A) / (cos A + 1 - sin A)

= (cos A + 1 - sin A) (cos A + 1 - sin A) / ( ((cos A - 1 + sin A) (cos A + 1 - sin A))

= (cos A + 1 - sin A)^2 / (cos^2 A - (1 - sin A)^2)

= (cos^2 A - 2 cos A (1 - sin A) + (1 - sin A)^2) / (cos^2 A - 1 + 2 sin A - sin^2 A)

= (cos^2 A - 2 cos A + 2 sin A cos A + 1 - 2 sin A + sin^2 A) / (1 - sin^2 A - 1 + 2 sin A - sin^2 A)

= (2 - 2 cos A + 2 sin A cos A - 2 sin A) / (2 sin A - 2 sin^2 A)

= 2 (1 - cos A) (1 - sin A) / (2 sin A (1 - sin A) )

= 1 - cos A / sin A

= RHS