Question : If $\operatorname{cosec} \theta-\cot \theta=\frac{7}{2}$, then the value of $\operatorname{cosec} \theta$ will be:
Option 1: $\frac{49}{28}$
Option 2: $\frac{21}{28}$
Option 3: $\frac{47}{28}$
Option 4: $\frac{53}{28}$
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Correct Answer: $\frac{53}{28}$
Solution :
$\operatorname{cosec}^2 \theta - \cot^2\theta = 1$
⇒$(\operatorname{cosec}\theta+\cot\theta)(\operatorname{cosec}\theta-\cot\theta) = 1$
Given, $\operatorname{cosec} \theta-\cot \theta=\frac{7}{2}$ ..........(i)
⇒$(\operatorname{cosec}\theta+\cot\theta)\frac{7}{2} = 1$
⇒$(\operatorname{cosec}\theta+\cot\theta) = \frac{2}{7}$ ............(ii)
Adding equation (i) and (ii), we get,
$2\operatorname{cosec}\theta = \frac{7}{2} + \frac{2}{7}$
$⇒2\operatorname{cosec}\theta = \frac{49+4}{14}$
$⇒\operatorname{cosec}\theta = \frac{53}{28}$
Hence, the correct answer is $\frac{53}{28}$.
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