Question : If $\operatorname{cosec} \theta+\cot \theta=2$, then what is the value of $\operatorname{cosec} \theta$?
Option 1: $\frac{5}{4}$
Option 2: $\sqrt{2}$
Option 3: $\sqrt{5}$
Option 4: $\frac{3}{2}$
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Correct Answer: $\frac{5}{4}$
Solution : We know that, $\operatorname{cosec}^2\theta-\operatorname{cot}^2\theta=(\operatorname{cosec}\theta-\operatorname{cot}\theta)(\operatorname{cosec}\theta+\operatorname{cot}\theta)=1$ $⇒(\operatorname{cosec}\theta-\operatorname{cot}\theta)=\frac{1}{2}$ ----------------(1) Given, $\operatorname{cosec} \theta+\cot \theta=2$ ------------------(2) Solving the above equations, we get, $\operatorname{cosec} \theta=\frac{5}{4}$ Hence, the correct answer is $\frac{5}{4}$.
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