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Let R be a relation defined on the open interval (0, pi/2) by R = {(a, b): cosec²(a) - cot²(b) = 1}. Then R is:

1. Reflexive and symmetric but not transitive
2. Reflexive and transitive but not symmetric
3. Symmetric and transitive but not reflexive
4. An equivalence relation

Correct answer: An equivalence relation

Solution

cosec²(a) - cot²(b) = 1 and cosec²(a) - cot²(a) = 1, so cot²(a) = cot²(b). Since both a, b in (0, pi/2), cot is strictly positive and injective, giving a = b. R is the identity relation on (0, pi/2), which is reflexive, symmetric, and transitive — an equivalence relation.

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