If (tan⁻¹ x)² + (cot⁻¹ x)² = 5pi²/8, then x equals
-1
0
1
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.