Question

CotA + cosecA -1 / CotA - cosecA + 1 = 1+cosA / 1+sinA

Answer 1

Firstly, You question is incorrect.. Instead of 1+CosA/1+SinA,it will be 1+CosA/SinA..

Solution:-
CosecA + CotA - 1 / CotA - CosecA + 1 
We know that,Cosec ²A - Cot ²A = 1 
Substituting this in the numerator,
CosecA + CotA -(Cosec ²A - Cot ²A) / (Cot A - CosecA + 1) 
x²-y²= (x+y)(x-y) 
Cosec A + CotA - (CosecA + CotA) (CosecA - CotA) / (CotA - CosecA + 1) 
Taking common,
(CosecA + CotA)(1-CosecA + CotA) / (CotA - CosecA + 1) 
Cancelling like terms in numerator and denominator,
We are left with cosec A + cot A 
= 1/sin A + cos A/sin A 
= (1+cos A) / sinA [Proved]

Answer 2

LHS

$\frac{cotA + cosecA - 1}{cotA - cosecA + 1} \\ \\ = \frac{(cotA + cosecA) - ( {cosec}^{2}A - {cot}^{2} A)}{cotA - cosecA + 1} \\ \\ = \frac{(cosecA + cotA)(1 - cosecA + cotA) }{cotA - cosecA + 1}$

$= cosecA + cotA \\ \\ = \frac{1 + cosA}{sinA}$

RHS