If and are real numbers

MATHEMATICS · Subjective · CBSE · Mathematics · Determinants · Determinants

Problem

If α,β\alpha, \beta and γ\gamma are real numbers, without expanding at any stage, prove that ∣1cos⁡(β−α)cos⁡(γ−α)cos⁡(α−β)1cos⁡(γ−β)cos⁡(α−γ)cos⁡(β−γ)1∣=0.\begin{vmatrix} 1 & \cos(\beta - \alpha) & \cos(\gamma - \alpha) \\ \cos(\alpha - \beta) & 1 & \cos(\gamma - \beta) \\ \cos(\alpha - \gamma) & \cos(\beta - \gamma) & 1 \end{vmatrix} = 0.

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