|(sin^223°,sin^267°,cos180°)(-sin^267°,-sin^223°,cos^2180°)(cos180°,sin^223°,sin^267°)|

Question

Without expanding, show that the value of each of the following determinants is zero :

\(\begin{vmatrix} sin^223° & sin^267° & cos180° \\[0.3em] -sin^267° & -sin^223° & cos^2180° \\[0.3em] cos180°& sin^223° & sin^267° \end{vmatrix}\)

Answer 1

Let Δ = \(\begin{vmatrix} sin^223° & sin^267° & cos180° \\[0.3em] -sin^267° & -sin^223° & cos^2180° \\[0.3em] cos180°& sin^223° & sin^267° \end{vmatrix}\)

Applying C₁→C₁ + C₂, we get

Therefore, 

As C₁ = C₃ determinant is zero.