Without expanding, show that the value of each of the following determinants is zero :|(sinα,cosα,cos(α+δ)),(sinβ,cosβ,cos(β+δ))(sinγ,cosγ,cos(γ+δ))|

Question

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

\(\begin{vmatrix} sin\,\alpha & cos\,\alpha & cos(\alpha+\delta) \\[0.3em] sin\,\beta & cos\,\beta & cos(\beta+\delta) \\[0.3em] sin\,\gamma & cos\,\gamma & cos(\gamma+\delta) \end{vmatrix}\)

Answer 1

Let Δ = \(\begin{vmatrix} sin\,\alpha & cos\,\alpha & cos(\alpha+\delta) \\[0.3em] sin\,\beta & cos\,\beta & cos(\beta+\delta) \\[0.3em] sin\,\gamma & cos\,\gamma & cos(\gamma+\delta) \end{vmatrix}\)

Multiplying C₁ with sin δ, C₂ with cos δ, we get

As C₂ = C₃ hence determinant is zero.