<p>If \(\cot^{-1} x + \cot^{-1} y + \cot^{-1} z = \dfrac{\pi}{2}\), then \(x + y + z\) equals</p>
Step-by-Step Solution
Key Concept: When cot⁻¹ x + cot⁻¹ y + cot⁻¹ z = π/2, convert using cot⁻¹ a = tan⁻¹(1/a) and apply the tangent addition formula for three angles whose sum is π/2, which means their complementary angles sum to π.
<p><strong>Step 1:</strong> Given: cot⁻¹ x + cot⁻¹ y + cot⁻¹ z = π/2</p><p><strong>Step 2:</strong> Let cot⁻¹ x = A, cot⁻¹ y = B, cot⁻¹ z = C, so A + B + C = π/2</p><p><strong>Step 3:</strong> This means cot A = x, cot B = y, cot C = z, and A + B + C = π/2</p><p><strong>Step 4:</strong> From A + B + C = π/2, we get A + B = π/2 - C</p><p><strong>Step 5:</strong> Taking tangent: tan(A + B) = tan(π/2 - C) = cot C = z</p><p><strong>Step 6:</strong> Using tan(A+B) formula: (tan A + tan B)/(1 - tan A tan B) = z</p><p><strong>Step 7:</strong> Since cot A = x, tan A = 1/x. Similarly, tan B = 1/y, tan C = 1/z</p><p><strong>Step 8:</strong> Substituting: (1/x + 1/y)/(1 - 1/(xy)) = 1/z</p><p><strong>Step 9:</strong> Simplifying: ((x+y)/(xy))/((xy-1)/(xy)) = 1/z, giving (x+y)/(xy-1) = 1/z</p><p><strong>Step 10:</strong> Cross multiply: z(x+y) = xy - 1, so xy + yz + zx = xy - 1</p><p><strong>Step 11:</strong> Therefore: <strong>x + y + z = xy + yz + zx</strong></p>
Correct Answer: A