Let α, β, γ be three real numbers and $$A=1cos$$⁡$$($$β−α$$)$$$$cos$$⁡$$($$γ−α$$)$$$$cos$$⁡$$($$α−β$$)1cos$$⁡$$($$γ−β$$)$$$$cos$$⁡$$($$α−γ$$)$$$$cos$$⁡$$($$β−γ$$)1$$ then

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Let α, β, γ be three real numbers and $$A=1cos$$⁡$$($$β−α$$)$$$$cos$$⁡$$($$γ−α$$)$$$$cos$$⁡$$($$α−β$$)1cos$$⁡$$($$γ−β$$)$$$$cos$$⁡$$($$α−γ$$)$$$$cos$$⁡$$($$β−γ$$)1$$ then

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|A|$$=1cos$$⁡$$($$β−α$$)$$$$cos$$⁡$$($$γ−α$$)$$$$cos$$⁡$$($$α−β$$)1cos$$⁡$$($$γ−β$$)$$$$cos$$⁡$$($$α−γ$$)$$$$cos$$⁡$$($$β−γ$$)1=$$$$cos$$⁡α$$sin$$⁡α$$0cos$$⁡β$$sin$$⁡β$$0cos$$⁡γ$$cos$$⁡γ$$0cos$$⁡α$$sin$$⁡α$$0cos$$⁡β$$sin$$⁡β$$0cos$$⁡γ$$sin$$⁡γ$$0=0$$⇒  A is singular.

Let $α, β, γ$ be three real numbers and $A = [\begin{array}{ccc} 1 & \cos (β - α) & \cos (γ - α) \\ \cos (α - β) & 1 & \cos (γ - β) \\ \cos (α - γ) & \cos (β - γ) & 1 \end{array}]$ then

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Let α, β, γ be three real numbers and A=1cos⁡(β−α)cos⁡(γ−α)cos⁡(α−β)1cos⁡(γ−β)cos⁡(α−γ)cos⁡(β−γ)1 then

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Let $α, β, γ$ be three real numbers and $A = [\begin{array}{ccc} 1 & \cos (β - α) & \cos (γ - α) \\ \cos (α - β) & 1 & \cos (γ - β) \\ \cos (α - γ) & \cos (β - γ) & 1 \end{array}]$ then