If \text{cosec } \theta-\cot \theta=m, then what is \text{cosec } \theta equal to?
- A. m+\frac{1}{m}
- B. m-\frac{1}{m}
- C. \frac{m}{2}+\frac{2}{m}
- D. \frac{m}{2}+\frac{1}{2m} ✓
Correct Answer: D. \frac{m}{2}+\frac{1}{2m}
Explanation
Using the identity \text{cosec}^2 \theta - \cot^2 \theta = 1, we get (\text{cosec } \theta - \cot \theta)(\text{cosec } \theta + \cot \theta) = 1. Since \text{cosec } \theta - \cot \theta = m, \text{cosec } \theta + \cot \theta = \frac{1}{m}. Adding the two equations gives 2\text{cosec } \theta = m + \frac{1}{m} \implies \text{cosec } \theta = \frac{m}{2} + \frac{1}{2m}.
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