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If the inequality |cos⁻¹ ((1-x²)/(1+x²)) |<(π)/(3) holds, then which of the following is true?

1. x ∈ [-(1)/(√(3)), (1)/(√(3)) ]
2. x ∈ (-(1)/(√(3)), (1)/(√(3)))
3. x ∈ (0, (1)/(√(3)))
4. None of these

Correct answer: x ∈ (-(1)/(√(3)), (1)/(√(3)))

Solution

Using arccos((1-x^2)/(1+x^2)) = 2 arctan|x|, the condition 2 arctan|x| < pi/3 gives arctan|x| < pi/6, so |x| < tan(pi/6) = 1/sqrt(3). Since the bound is strict, x lies in the open interval (-1/sqrt(3), 1/sqrt(3)).

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