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

(N/A) We start with the Left Hand Side $(LHS)$:
$LHS = \frac{\cot A - \cos A}{\cot A + \cos A}$
Substitute $\cot A = \frac{\cos A}{\sin A}$:
$LHS = \frac{\frac{\cos A}{\sin A} - \cos A}{\frac{\cos A}{\sin A} + \cos A}$
Factor out $\cos A$ from the numerator and the denominator:
$LHS = \frac{\cos A \left( \frac{1}{\sin A} - 1 \right)}{\cos A \left( \frac{1}{\sin A} + 1 \right)}$
Cancel $\cos A$ from both:
$LHS = \frac{\frac{1}{\sin A} - 1}{\frac{1}{\sin A} + 1}$
Since $\operatorname{cosec} A = \frac{1}{\sin A}$,we substitute it:
$LHS = \frac{\operatorname{cosec} A - 1}{\operatorname{cosec} A + 1} = RHS$
Hence,the identity is proved.