<p>Solve: tan⁻¹((x+1)/(x-1)) + tan⁻¹((x-1)/x) = tan⁻¹(-7)</p>
Step-by-Step Solution
Key Concept: Apply the addition formula for inverse tangent functions to combine the two terms, then equate arguments and solve the resulting quadratic equation.
<p><strong>Step 1:</strong> Use the addition formula: tan⁻¹(a) + tan⁻¹(b) = tan⁻¹((a+b)/(1-ab))</p><p><strong>Step 2:</strong> Let a = (x+1)/(x-1) and b = (x-1)/x</p><p><strong>Step 3:</strong> a + b = (x+1)/(x-1) + (x-1)/x = (x² + x + x² - 2x + 1)/(x(x-1)) = (2x² - x + 1)/(x² - x)</p><p><strong>Step 4:</strong> 1 - ab = 1 - [(x+1)/(x-1)]·[(x-1)/x] = 1 - (x+1)/x = (x - x - 1)/x = -1/x</p><p><strong>Step 5:</strong> Therefore: tan⁻¹((2x² - x + 1)/(1 - x)) = tan⁻¹(-7)</p><p><strong>Step 6:</strong> (2x² - x + 1)/(1 - x) = -7</p><p><strong>Step 7:</strong> 2x² - x + 1 = -7(1 - x) = -7 + 7x</p><p><strong>Step 8:</strong> 2x² - 8x + 8 = 0 ⟹ x² - 4x + 4 = 0</p><p><strong>Step 9:</strong> (x - 2)² = 0 ⟹ x = 2</p><p>∴ Answer is 2</p>
Correct Answer: 2