Prove that (Cosec a - Sin A)( Sec a - Cos A) Sec2 a = Tan A.

Prove that (cosec A - sin A)( sec A - cos A) sec² A = tan A.

Sum

LHS = (cosec A - sin A)(sec A - cos A). sec²A

= `(1/sin A - sin A).(1/cos A - cos A). 1/cos^2 A`

= `(1- sin^2A)/sin A.(1- cos^2 A)/(cos A) xx 1/cos^2 A`

= `cos^2 A/sin A xx sin^2 A/cos A xx 1/cos^2 A ....[ ∵ ( 1 - sin^2 A) = cos^2 A, 1 - cos^2 A = sin^2 A]`

= `sin A/cos A = tan A`
= RHS
Hence proved.

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