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Solve for x: arctan((x - 1)/(x - 2)) + arctan((x + 1)/(x + 2)) = pi/4.

1. x = 1/sqrt2
2. x = -1/sqrt2
3. x = sqrt2
4. x = -sqrt2

Correct answer: x = 1/sqrt2

Solution

Let a = (x-1)/(x-2), b = (x+1)/(x+2). Then (a + b)/(1 - ab) = tan(pi/4) = 1. Computing a + b = (2x² - 4)/((x-2)(x+2)) = (2x² - 4)/(x² - 4), and ab = (x² - 1)/(x² - 4). So 1 - ab = (x² - 4 - x² + 1)/(x² - 4) = -3/(x² - 4). Setting (a+b)/(1-ab) = (2x² - 4)/(-3) = 1 gives 2x² - 4 = -3, so 2x² = 1, x = +/- 1/sqrt2. Checking the principal-value/branch condition, x = 1/sqrt2 is the valid root.

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