Question : The value of $\frac{\sin A}{\cot A+\operatorname{cosec} A}-\frac{\sin A}{\cot A-\operatorname{cosec} A}+1$ is:
Option 1: $\frac{1}{2}$
Option 2: $3$
Option 3: $0$
Option 4: $2$
Correct Answer: $3$
Solution :
$\frac{\sin A}{\cot A+\operatorname{cosec} A}-\frac{\sin A}{\cot A-\operatorname{cosec} A}+1$
$=\frac{\sin A}{\frac{\cos A}{\sin A}+\frac{1}{\sin A}}-\frac{\sin A}{\frac{\cos A}{\sin A}-\frac{1}{\sin A}}+1$
$=\frac{\sin^2 A}{\cos A+1}-\frac{\sin^2 A}{\cos A-1}+1$
$=\frac{\sin^2 A}{1+\cos A}+\frac{\sin^2 A}{1-\cos A}+1$
$=\sin^2 A[\frac{1}{1+\cos A}+\frac{1}{1-\cos A}]+1$
$=\sin^2 A[\frac{2}{1-\cos ^2A}]+1$
$=\sin^2 A[\frac{2}{\sin ^2A}]+1$
$=2+1$
$=3$
Hence, the correct answer is $3$.
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