191
Prove that $\frac{\cos A + \sin A - 1}{\cos A - \sin A + 1} = \csc A - \cot A$
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$LHS = \frac{\cos A + \sin A - 1}{\cos A - \sin A + 1} = \frac{\cot A + 1 - \csc A}{\cot A - 1 + \csc A}$
$= \frac{\cot A - \csc A + \csc^2 A - \cot^2 A}{\cot A - 1 + \csc A}$
$= \frac{(\csc A - \cot A)(-1 + \csc A + \cot A)}{\cot A - 1 + \csc A}$
$= \csc A - \cot A = RHS$
$= \frac{\cot A - \csc A + \csc^2 A - \cot^2 A}{\cot A - 1 + \csc A}$
$= \frac{(\csc A - \cot A)(-1 + \csc A + \cot A)}{\cot A - 1 + \csc A}$
$= \csc A - \cot A = RHS$