189
Prove that $\frac{\cos A + \sin A-1}{\cos A-\sin A +1} = \text{cosec } A - \cot A$
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LHS = $\frac{\cos A+\sin A-1}{\cos A-\sin A+1}$
$= \frac{\cot A+1-\text{cosec } A}{\cot A-1+\text{cosec } A}$
$= \frac{\cot A-\text{cosec } A+\text{cosec}^2A- \cot^2 A}{\cot A-1+\text{cosec } A}$
$= \frac{(\text{cosec } A-\cot A)(-1+\text{cosec } A+\cot A)}{\cot A-1+\text{cosec } A}$
$= \text{cosec } A - \cot A = \text{RHS}$
$= \frac{\cot A+1-\text{cosec } A}{\cot A-1+\text{cosec } A}$
$= \frac{\cot A-\text{cosec } A+\text{cosec}^2A- \cot^2 A}{\cot A-1+\text{cosec } A}$
$= \frac{(\text{cosec } A-\cot A)(-1+\text{cosec } A+\cot A)}{\cot A-1+\text{cosec } A}$
$= \text{cosec } A - \cot A = \text{RHS}$