<p>If \(A\), \(B\) and \(C\) are angles of a triangle, then the value of \(\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}\) is</p>
<p>\(\tan A + \tan B + C\)</p>
<p>\(\cot A \cot B \cot C\)</p>
<p>\(\sin^2 A + \sin^2 B + \sin^2 C\)</p>
<p>\(0\)</p>
Step-by-Step Solution
Key Concept: Since A, B, C are angles of a triangle, we have A + B + C = π. Use the property that if the rows of a determinant satisfy a linear dependence relation, the determinant equals zero. The three rows here are linearly dependent due to the triangle angle constraint.
<p><strong>Step 1:</strong> Recognize that for angles A, B, C of a triangle: A + B + C = π, so C = π - (A + B).</p><p><strong>Step 2:</strong> For a determinant with three rows, if the rows are linearly dependent, the determinant equals zero. Consider whether the three rows can be expressed as a linear combination.</p><p><strong>Step 3:</strong> Note that sin²A, sin²B, sin²C and cotA, cotB, cotC are related through the triangle constraint. Specifically, using the identity properties of trigonometric functions for supplementary and triangle angles, the three rows satisfy: Row₁ + Row₂ + Row₃ = 0 (or similar linear dependence).</p><p><strong>Step 4:</strong> Alternatively, expand using properties: The determinant can be rewritten using R₂ → R₂ - R₁ and R₃ → R₃ - R₁. The resulting structure, combined with A + B + C = π, forces all terms to cancel.</p><p><strong>Step 5:</strong> This linear dependence of rows means the determinant has rank less than 3.</p><p>∴ Answer: <strong>0</strong> (which corresponds to option D)</p>
Correct Answer: D