Prove that the determinant is independent of θ.

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Solution

$Δ = | \begin{matrix} x & \sin θ & \cos θ \\ - \sin θ & - x & 1 \\ \cos θ & 1 & x \end{matrix} | = x (- x^{2} - 1) - \sin θ (- x \sin θ - \cos θ) + \cos θ (- \sin θ + x \cos θ) = - x^{3} - x + x \sin^{2} θ + \sin θ \cos θ - \sin θ \cos θ + x \cos^{2} θ = - x^{3} - x + x (\sin^{2} θ + \cos^{2} θ) = {- x}^{3} - x + x = - x^{3} (Independent of θ)$

Hence, Δ is independent of θ.

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