If α,β,γ are three real numbers such that α$$+$$β$$+$$γ$$=0$$, then Δ$$=1cos$$⁡γ$$cos$$⁡β$$cos$$⁡γ$$1cos$$⁡α$$cos$$⁡β$$cos$$⁡α$$1$$ equals

Question

If α,β,γ are three real numbers such that α$$+$$β$$+$$γ$$=0$$, then Δ$$=1cos$$⁡γ$$cos$$⁡β$$cos$$⁡γ$$1cos$$⁡α$$cos$$⁡β$$cos$$⁡α$$1$$ equals

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Let  A,B and C be three real numbers such that α$$=B$$−C,β$$=C$$−A and γ$$=A$$−B, clearly α$$+$$β$$+$$γ$$=0$$. We have $$=cos2$$⁡$$A+sin2$$⁡Acos⁡Acos⁡$$B+$$$$sin$$⁡Asin⁡Bcos⁡Acos⁡$$B+$$$$sin$$⁡Asin⁡$$Bcos2$$⁡$$B+sin2$$⁡Bcos⁡Ccos⁡$$A+$$$$sin$$⁡Csin⁡Acos⁡Bcos⁡$$C+$$$$sin$$⁡Bsin⁡C    $$cos$$⁡Ccos⁡$$A+$$$$sin$$⁡Asin⁡Ccos⁡Bcos⁡$$C+$$$$sin$$⁡Bsin⁡$$Ccos2$$⁡$$C+sin2$$⁡C               $$=$$$$cos$$⁡A    $$sin$$⁡A    $$0cos$$⁡B    $$sin$$⁡B    $$0cos$$⁡C    $$sin$$⁡C    $$0cos$$⁡Asin⁡$$A0cos$$⁡Bsin⁡$$B0cos$$⁡Csin⁡$$C0=(0)(0)=0$$.

If $α, β, γ$ are three real numbers such that $α + β + γ = 0,$ then $Δ = |\begin{array}{ccc} 1 & \cos γ & \cos β \\ \cos γ & 1 & \cos α \\ \cos β & \cos α & 1 \end{array}|$ equals

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If α,β,γ are three real numbers such that α+β+γ=0, then Δ=1cos⁡γcos⁡βcos⁡γ1cos⁡αcos⁡βcos⁡α1 equals

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If $α, β, γ$ are three real numbers such that $α + β + γ = 0,$ then $Δ = |\begin{array}{ccc} 1 & \cos γ & \cos β \\ \cos γ & 1 & \cos α \\ \cos β & \cos α & 1 \end{array}|$ equals