Question : If $(\operatorname{cosec} \theta-\cot \theta) = \frac{7}{2}$, the value of $\operatorname{cosec} \theta$ is:
Option 1: $\frac{47}{28}$
Option 2: $\frac{51}{28}$
Option 3: $\frac{53}{28}$
Option 4: $\frac{49}{28}$
Correct Answer: $\frac{53}{28}$
Solution :
Given: $(\operatorname{cosec} \theta-\cot \theta) = \frac{7}{2}$ (equation 1)
We know the trigonometric identity, $(\operatorname{cosec} \theta-\cot \theta)(\operatorname{cosec} \theta+\cot \theta)=1$
Substitute the value from equation 1 into the above identity, and we get,
⇒ $\frac{7}{2}\times(\operatorname{cosec} \theta+\cot \theta) = 1$
⇒ $(\operatorname{cosec} \theta+\cot \theta) = \frac{2}{7}$ (equation 2)
Adding equations 1 and 2, we get,
$(\operatorname{cosec} \theta–\cot \theta) + (\operatorname{cosec} \theta+\cot \theta) = \frac{7}{2} + \frac{2}{7}$
⇒ $2\times \operatorname{cosec} \theta = \frac{53}{14}$
$\therefore \operatorname{cosec} \theta = \frac{53}{28}$
Hence, the correct answer is $\frac{53}{28}$.
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