Prove that $\frac{\cos A + \sin A - 1}{\cos A - \sin A + 1} = \cosec A - \cot A$.

Question

Accepted Answer

Step 1: LHS $= \frac{\cos A + \sin A - 1}{\cos A - \sin A + 1}$. Dividing numerator and denominator by $\sin A$:
$= \frac{\cot A + 1 - \cosec A}{\cot A - 1 + \cosec A}$
Step 2: Using $\cosec^2 A - \cot^2 A = 1$ in the numerator:
$= \frac{\cot A - \cosec A + (\cosec A - \cot A)(\cosec A + \cot A)}{\cot A - 1 + \cosec A}$
Step 3: $= \frac{(\cosec A - \cot A)(\cosec A + \cot A - 1)}{\cosec A + \cot A - 1} = \cosec A - \cot A = $ RHS

Question

Solution