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

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Solution

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

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Prove: $\frac{cot A + c o s e c A - 1}{cot A - c o s e c A + 1} = \frac{1 + cos A}{sin A}$

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