Question

By using properties of determinants, prove that :

\(\begin{vmatrix} 1+sin^ 2x & cos^2x & 4\,sin\,2x \\ sin^2x & 1+cos^2x & 4\,sin\,2x \\ sin^2x & cos^2x & 1+4\,sin\,2x \end{vmatrix}\) = 2 + 4 sin 2x

Answer 1

L.H.S = \(\begin{vmatrix} 1+sin^ 2x & cos^2x & 4\,sin\,2x \\ sin^2x & 1+cos^2x & 4\,sin\,2x \\ sin^2x & cos^2x & 1+4\,sin\,2x \end{vmatrix}\)

Applying C₁ → C₂,

Applying R₂ → R₂ - R₁, R₃ → R₃ - R₁ ,

= 2(1 - 0) - 1 (0 - 4 sin 2x)

= 2 + 4 sin 2x = R.H.S