Question : If $\operatorname{cosec} \theta+ \cot \theta=\frac{3}{2}$, what is the value of $\operatorname{cosec} \theta$?
Option 1: $\frac{13}{12}$
Option 2: $\frac{3}{2}$
Option 3: $\frac{11}{12}$
Option 4: $\frac{9}{13}$
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Correct Answer: $\frac{13}{12}$
Solution :
Given: $\operatorname{cosec}\ \theta+\cot \theta=\frac{3}{2}$ -----------------(1)
Since we know that $(\operatorname{cosec} \ \theta+\cot \theta)(\operatorname{cosec}\ \theta-\cot \theta)=1$
Substitute the given value in the above identity and we get,
$⇒\frac{3}{2}×(\operatorname{cosec} \ \theta-\cot \theta)=1$
$⇒\operatorname{cosec} \ \theta-\cot \theta=\frac{2}{3}$-------------------------(2)
Adding equation (1) and equation (2), we get,
$\operatorname{cosec}\ \theta+\cot \theta+\operatorname{cosec}\ \theta-\cot \theta=\frac{9+4}{6}$
$⇒2\operatorname{cosec}\ \theta=\frac{13}{6}$
$⇒\operatorname{cosec}\ \theta=\frac{13}{12}$
Hence, the correct answer is $\frac{13}{12}$.
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