Question:
$\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$
Solution:
$\frac{\cot A-\cos A}{\cot A+\cos A}$
$=\frac{\frac{\cos A}{\sin A}-\cos A}{\frac{\cos A}{\sin A}+\cos A}$
$=\frac{\cos A\left(\frac{1}{\sin A}-1\right)}{\cos A\left(\frac{1}{\sin A}+1\right)}$
$=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1} \quad\left(\operatorname{cosec} A=\frac{1}{\sin A}\right)$
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