Question

cos theta-sin theta+1÷cos theta+sin theta+1=cosec theta+cot theta

Answer 1

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

Answer 2

hope this answer help u !