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 $\frac{13}{12}$.