Cos theta minus sin theta + 1 upon cos theta + sin theta minus 1 is equal to cosec theta + cot theta

Question

(Cos θ - Sin θ + 1)/ (Cos θ + Sin θ - 1) =  Cosec θ + Cot θ

Answer 1

Let θ = A

Now,

Answer 2

LHS 

(cos∅ - sin∅ + 1)/(cos∅ + sin∅-1)
dividing sin∅ both Numerator and denominator.
= (cos∅/sin∅ - sin∅/sin∅ + 1/sin∅)/(cos∅/sin∅ + sin∅/sin∅ - 1/sin∅)
= (cot∅ - 1 + cosec∅)/(cot∅ + 1 - cosec∅)

now, put 1 = cosec²∅ - cot²∅ in numerator

= {cot∅ + cosec∅ - (cosec²∅-cot²∅)}/(cot∅-cosec∅ +1)
= (cosec∅+cot∅)(1 - cosec∅ + cot∅)/(cot∅-cosec∅+1)
= cosec∅ + cot∅ = RHS

Hence Proved.