We have to prove that (sin A - cos A + 1)/(sin A + cos A - 1) = cos A/(1 - sin A)
Start from the left hand side
(sin A - cos A + 1)/(sin A + cos A - 1)
=> (sin A - cos A +...
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We have to prove that (sin A - cos A + 1)/(sin A + cos A - 1) = cos A/(1 - sin A)
Start from the left hand side
(sin A - cos A + 1)/(sin A + cos A - 1)
=> (sin A - cos A + 1)(sin A - cos A -1)/(sin A + cos A - 1)(sin A - cos A -1)
=> ((sin A - cos A)^2 - 1)/((sin A - 1)^2 - (cos A)^2)
=> ((sin A - cos A)^2 - 1)/((sin A)^2 - 2*sin A + 1 - (cos A)^2)
=> ((sin A - cos A)^2 - 1)/((sin A)^2 - 2*sin A + (sin A)^2)
=> ((sin A)^2 + (cos A)^2 - 2*sin A*cos A - 1)/((sin A)^2 - 2*sin A + (sin A)^2)
=> (1- 2*sin A*cos A - 1)/((sin A)^2 - 2*sin A + (sin A)^2)
=> (-2*sin A*cos A)/(2*(sin A)^2 - 2*sin A)
=> (-cos A)/(sin A - 1)
=> cos A/(1 - sin A)
which is the right hand side
This proves:(sin A - cos A + 1)/(sin A + cos A - 1) = cos A/(1 - sin A)