Question

((cotA+cosecA-1)/(cotA-cosecA+1))=(1+cosA)/(sinA)

Answer 1

Cosec A + cot A - 1 / cot A - cosec A + 1 
we know that,cosec ² A - cot ² A = 1 
substituting this in the numerator 
cosec A + cot A -(cosec ² A - cot ² A) / (cot A - cosec A + 1) 
x²-y²= (x+y)(x-y) 
cosec A + cot A - (cosec A + cot A) (cosec A - cot A) / (cot A - cosec A + 1) 
taking common 
(cosec A + cot A)(1-cosec A + cot A) / (cot A - cosec A + 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) / sin A

Answer 2

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

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