Question : The value of $\frac{1}{\sin \theta}-\frac{\cot ^2 \theta}{1+\operatorname{cosec} \theta}$ is:
Option 1: 2
Option 2: 1
Option 3: –1
Option 4: 0
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Correct Answer: 1
Solution :
Given:
$\frac{1}{\sin \theta}-\frac{\cot ^2 \theta}{1+\operatorname{cosec} \theta}$
= $\operatorname {cosec} \theta - \frac{\operatorname{cosec}^2\theta-1}{1+\operatorname{cosec} \theta}$
= $\operatorname {cosec}\theta-\frac{(\operatorname{cosec}\theta-1)(\operatorname{cosec}\theta+1)}{1+\operatorname{cosec} \theta}$
= $\operatorname{cosec}\theta-\operatorname{cosec}\theta+1$
= $1$
Hence, the correct answer is 1.
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