Prove that the determinant |(x,sin θ,cos θ),(-sin θ,-x,1),(cos θ,1

1.	Prove that the determinant \|(x,sin θ,cos θ),(-sin θ,-x,1),(cos θ,1,2)\| is independent of θ.
Answer» Expending along the first row, we get Δ = x(-x² - 1) - sin θ(-x sin θ - cos θ) + cos θ(-sin θ + x cos θ) = - x³ + x + x sin² θ + sin θ.cos θ - sin θ.cos θ + x cos² θ = x³ + x + x(sin² θ + cos²θ) = - x³ + 2x Hence, given determinant is independent of θ.

Prove that the determinant |(x,sin θ,cos θ),(-sin θ,-x,1),(cos θ,1,2)| is independent of θ.

Answer»

Expending along the first row, we get

Δ = x(-x² - 1) - sin θ(-x sin θ - cos θ) + cos θ(-sin θ + x cos θ)

= - x³ + x + x sin² θ + sin θ.cos θ - sin θ.cos θ + x cos² θ

= x³ + x + x(sin² θ + cos²θ) = - x³ + 2x

Hence, given determinant is independent of θ.