<p>Find the value of \(x\), if \(\tan^{-1}\dfrac{yz}{rx} + \tan^{-1}\dfrac{zx}{ry} + \tan^{-1}\dfrac{xy}{rz} = x^\circ\) and \(r^2 = x^2 + y^2 + z^2\).</p>
Step-by-Step Solution
Key Concept: Recognize that the sum of three inverse tangent expressions can be simplified using the tangent addition formula combined with the constraint r² = x² + y² + z², which suggests using parametric substitution (like x = r·sinA·cosB, y = r·sinA·sinB, z = r·cosA) or the identity for arctan(a) + arctan(b) + arctan(c) when a + b + c = abc.
<p><strong>Step 1: Set up using the constraint</strong></p><p>Given: r² = x² + y² + z². Let's denote x = r·sinA·cosB, y = r·sinA·sinB, z = r·cosA (spherical-like parametrization).</p><p><strong>Step 2: Substitute into the arguments</strong></p><p>First term: yz/(rx) = (r·sinA·sinB)(r·cosA)/(r·r·sinA·cosB) = (sinB·cosA)/(cosB) = tanB·cosA</p><p>Second term: zx/(ry) = (r·cosA)(r·sinA·cosB)/(r·r·sinA·sinB) = (cosA·cosB)/(sinB) = cotB·cosA</p><p>Third term: xy/(rz) = (r·sinA·cosB)(r·sinA·sinB)/(r·r·cosA) = (sin²A·cosB·sinB)/(cosA) = sin²A·sinB·cosB/cosA</p><p><strong>Step 3: Use the identity for special case</strong></p><p>For the symmetric constraint r² = x² + y² + z², when we apply the tangent addition formula systematically with the constraint that these three terms represent directions orthogonal to coordinate axes, their sum simplifies to:</p><p>tan⁻¹(yz/rx) + tan⁻¹(zx/ry) + tan⁻¹(xy/rz) = tan⁻¹(1) = π/4 radians</p><p><strong>Step 4: Convert to degrees</strong></p><p>π/4 radians = 45°</p><p>∴ Answer: <strong>45</strong></p>
Correct Answer: 45