Question

cos A- sin A+1/ cos A+sinA-1 =cosec A+cot A​

Answer 1

(cos A - sin A + 1) / (cos A + sin A - 1)

divide numerator and denominator by sin A

(cos A/sin A - sin A/sin A + 1/sin A) / (cos A/sin A + sin A/sin A - 1/sin A)

(cot A - 1 + cosec A) / (cot A + 1 - cosec A)

(cosec A + cot A - 1) / (cot A + 1 - cosec A)

Now replace the '1' in numerator by :-

cosec^2 A = 1 + cot^2 A

1 = cosec^2 A - cot^2 A

[cosec A + cot A - (cosec^2 A - cot^2 A)] / (cot A + 1 - cosec A)

Now,

cosec^2 A - cot^2 A =  (cosec A + cot A)(cosec A - cot A)

[(cosec A + cot A) - (cosec A - cot A)(cosec A + cot A)] / (cot A +1 - cosecA)

now take , (cosec A + cot A) common from numerator

(cosec A + cot A)[1 -(cosec A - cot A)] / (cot A +1 - cosecA)

(cosec A + cot A)[1 - cosec A + cot A)] / (cot A +1 - cosecA)

after cancel out

cosec A + cot A

PRETTY LONG........

HOPE IT HELPS