Cosa-sina+1/cosa+sina-1=coseca+cota=using the identity cosec2a=1+cot2a

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Solution

Start from LHS:
cosA-sinA+1/cosA+sinA-1

Divide numerator and denominator by sin A.
cot A - 1 + cosec A / cot A + 1 - cosec A

Now rearrange:
cot A + cosec A - 1/ cot A - cosec A + 1

Put 1 = cosec^2 A - cot^2 A in the numerator:
cot A + cosec A - (cosec^2 A - cot^2 A)/ cot A - cosec A + 1

Use the identity a^2 - b^2 = (a + b)(a - b):
(cot A + cosec A) - (cosec A - cot A)(cosec A + cot A) / (cot A - cosec A + 1)

Now take (cosec A + cot A) common in the numerator:
(cot Aa + cosec A)(1 - cosec A + cot A) / (cot A - cosec A + 1)
= cot A + cosec A
= RHS

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

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\frac{cos A + sin A - 1}{cos A - sin A + 1} = \frac{1}{cosec A + cot A}

{cosec}^{2} A - {cot}^{2} A = 1.

\frac{cos A - sin A + 1}{cos A + sin A - 1} = cosec A + cot A

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\frac{(c o s A - s i n A + 1)}{(c o s A + s i n A - 1)} = c o s e c A + c o t A

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