Prove that (cotQ+ cosecQ -1) / (cotQ – cosecQ +1) = (1-cosQ) / sinQ?

Here we have to prove (cotQ+ cosecQ -1) / (cotQ – cosecQ +1) = (1-cosQ) / sinQ we will start with LHS (Left hand side ie (cotQ+ cosecQ -1) / (cotQ – cosecQ +1) and prove that it is equal to (1-cosQ) / sinQ Hence we will get LHS = RHS

Proof

Lets start with LHS of the equations
LHS = (cotQ+ cosecQ -1) / (cotQ – cosecQ +1)

= (cosQ/ sinQ + 1/sinQ – sinQ / sinQ) / (cosQ / sinQ – 1 / sinQ + sinQ / sinQ)

= (cosQ+ 1 – sinQ) / (cosQ – 1 + sinQ) * (cosQ + 1 – sinQ) / (cosQ + 1 – sinQ)

= (cosQ + 1 – sinQ) (cosQ + 1 – sinQ) / ( ((cosQ – 1 + sinQ) (cosQ + 1 – sinQ))

= (cosQ + 1 – sinQ)² / (cos²Q – (1 – sinQ)²)

= (cos²Q – 2 cosQ (1 – sinQ) + (1 – sinQ)²) / (cos²Q – 1 + 2 sinQ – sin²Q)

= (cos²Q – 2 cosQ + 2 sinQ cosQ + 1 – 2 sinQ + sin²Q) / (1 – sin² Q – 1 + 2 sinQ – sin²Q)
= (2 – 2 cosQ + 2 sinQ cosQ – 2 sinQ) / (2 sinQ – 2 sin²Q)
= 2 (1 – cosQ) (1 – sinQ) / (2 sinQ (1 – sinQ) )
= 1 – cosQ / sinQ
= RHS

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