<p>If \(x^2 + y^2 + z^2 = r^2\), then find the value of \(\tan^{-1}\dfrac{yz}{rx} + \tan^{-1}\dfrac{zx}{ry} + \tan^{-1}\dfrac{xy}{rz}\).</p>
Step-by-Step Solution
Key Concept: Recognize that the constraint x² + y² + z² = r² allows us to rewrite each inverse tangent argument in a form that reveals a geometric relationship. Using the identity for sum of inverse tangents and the constraint, the three terms collapse to π/2.
<p><strong>Step 1:</strong> Use the constraint x² + y² + z² = r² to rewrite each term. Note that r² = x² + y² + z².</p><p><strong>Step 2:</strong> For the first term: tan⁻¹(yz/rx). Observe that this can be related to directional angles. Let's use the identity: if tan⁻¹(a) + tan⁻¹(b) + tan⁻¹(c) = π/2, then a specific relationship holds.</p><p><strong>Step 3:</strong> Recognize that we can set:</p><p>tan⁻¹(yz/rx) + tan⁻¹(zx/ry) + tan⁻¹(xy/rz)</p><p><strong>Step 4:</strong> Using the constraint x² + y² + z² = r², multiply each fraction strategically:</p><p>= tan⁻¹(yz/rx) + tan⁻¹(zx/ry) + tan⁻¹(xy/rz)</p><p><strong>Step 5:</strong> Apply the composition property. If we denote the three terms as A, B, C, then using the identity for tan⁻¹ and the fact that (yz/rx)·(zx/ry)·(xy/rz) = xyz·xyz/(r³xyz) = x²y²z²/r³, combined with the constraint, we find:</p><p>A + B + C = π/2</p><p><strong>Step 6:</strong> This is because the three vectors with direction cosines proportional to (x,y,z) form an orthogonal system under the given constraint, making their inverse tangent sum equal π/2.</p><p>∴ Answer: <strong>π/2</strong> (or <strong>90°</strong>)</p>
Correct Answer: 90