Question

Prove that the determinant

\(\begin{vmatrix} x &sin\theta & cos\theta \\[0.3em] -sin\theta& -x &1 \\[0.3em] 1 &1 &x \end{vmatrix}\) is independent of θ

Answer 1

(-xsinθ - cosθ) - cosθ(-sinθ + xcosθ)

= - x³ - x + xsin²θ + sinθcosθ = sinθcosθ + xcos²θ

= -x³ - x + x(sin²θ + cos²θ)

= -x³ - x + x X 1 = - x³

which is independent of θ