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If (tan⁻¹ x)² + (cot⁻¹ x)² = 5pi²/8, then x equals

1. -1
2. 0
3. 1
4. none of these

Correct answer: -1

Solution

With a = tan⁻¹ x and b = cot⁻¹ x, a + b = pi/2. Then a² + b² = (a+b)² - 2ab = pi²/4 - 2ab = 5pi²/8 gives ab = -3pi²/16. Solving the resulting quadratic gives a = -pi/4 (the value in tan⁻¹ range), so x = tan(-pi/4) = -1.

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