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Prove : Cos A - Sin A +1 / Cos A +Sin A -1 = Cosec A + Cot A using the identity cosec^{2}A = 1+cot^{2}A

Cot A - 1 + Cosec A / Cot A - Cosec A + 1

now putting 1= Cosec^2 - Cot ^2

= (Cot A + Cosec A) - (Cosec2 A - Cot2 A) / (Cot A - Cosec A + 1)

= (Cot A + Cosec A) [1 - Cosec A + Cot A) / (Cot A - Cosec A + 1)]

= (Cot A + Cosec A) RHS Proved