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Solve the following inequalities: (i) (arccot x)² - 5 arccot x + 6 > 0 (ii) arcsin x > arccos x (iii) tan²(arcsin x) > 1

1. (i) x < cot 3 or x > cot 2; (ii) x in (1/sqrt(2), 1]; (iii) x in (-1, -1/sqrt(2)) U (1/sqrt(2), 1)
2. (i) cot 3 < x < cot 2; (ii) x in [0, 1]; (iii) x in (-1/sqrt(2), 1/sqrt(2))
3. (i) all real x; (ii) x in (1/sqrt(2), 1]; (iii) x in (-1, 1)
4. (i) x > cot 2 only; (ii) x in (0, 1); (iii) |x| < 1/sqrt(2)

Correct answer: (i) x < cot 3 or x > cot 2; (ii) x in (1/sqrt(2), 1]; (iii) x in (-1, -1/sqrt(2)) U (1/sqrt(2), 1)

Solution

Each reduces to a basic inequality in the inverse-function variable, then is mapped back to x noting the monotonic direction of the inverse function and the domain [-1,1] where relevant.

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