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Find all x satisfying (arccot x)² - 7 (arccot x) + 10 > 0, where arccot has principal range (0, pi).

1. (-infinity, cot 5) U (cot 2, infinity)
2. (cot 5, cot 4)
3. (cot 2, infinity)
4. (-infinity, cot 5) U (cot 4, cot 2)

Correct answer: (cot 2, infinity)

Solution

Let y = arccot x in (0, pi). (y-2)(y-5) > 0 holds for y < 2 or y > 5. Since y < pi ≈ 3.14, y > 5 is impossible, so we need y < 2, i.e. 0 < arccot x < 2. arccot is decreasing, so arccot x < 2 <=> x > cot 2. Therefore x in (cot 2, infinity).

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