If A,B,C,P,Q,R∈R and Δ$$=$$$$cos$$⁡(A+P) $$cos$$⁡(A+Q) $$cos$$⁡(A+R)$$cos$$⁡(B+P) $$cos$$⁡(B+Q) $$cos$$⁡(B+R)$$cos$$⁡(C+P) $$cos$$⁡(C+Q) $$cos$$⁡(C+R)

Question

If A,B,C,P,Q,R∈R and Δ$$=$$$$cos$$⁡(A+P)    $$cos$$⁡(A+Q)    $$cos$$⁡(A+R)$$cos$$⁡(B+P)    $$cos$$⁡(B+Q)    $$cos$$⁡(B+R)$$cos$$⁡(C+P)    $$cos$$⁡(C+Q)    $$cos$$⁡(C+R)

Infinity Learn · Accepted Answer

WriteΔ$$=$$$$cos$$⁡Asin⁡$$A0cos$$⁡Bsin⁡$$B0cos$$⁡Csin⁡$$C0cos$$⁡P−$$sin$$⁡$$P0cos$$⁡Q−$$sin$$⁡$$Q0cos$$⁡R−$$sin$$⁡$$R0=0Alternate$$ SolutionWrite Δ$$=$$$$cos$$⁡PΔ$$1$$−$$sin$$⁡PΔ$$2write$$ Δ$$1=$$$$cos$$⁡A    $$cos$$⁡(A+Q)    $$cos$$⁡(A+R)$$cos$$⁡B    $$cos$$⁡(B+Q)    $$cos$$⁡(B+R)$$cos$$⁡C    $$cos$$⁡(C+Q)    $$cos$$⁡(C+R)and$$ Δ$$2=$$$$sin$$⁡A    $$cos$$⁡$$(A+Q)    $$cos$$⁡(A+R)$$sin$$⁡B    $$cos$$⁡(B+Q)    $$cos$$⁡(B+R)$$sin$$⁡C    $$cos$$⁡(C+Q)    $$cos$$⁡$$(C+R)In$$ Δ$$1$$,use $$C2$$→$$C2$$−$$cos$$⁡$$QC1C3$$→$$C3$$−$$cos$$⁡$$RC1$$Δ$$1=$$$$cos$$⁡A    $$sin$$⁡Asin⁡Q    $$sin$$⁡Asin⁡Rcos⁡B    $$sin$$⁡Bsin⁡Q    $$sin$$⁡Bsin⁡Rcos⁡C    $$sin$$⁡Csin⁡Q    $$sin$$⁡Csin⁡$$R=0$$                     ∵$$C2$$ and $$C3$$ are propotional Similarly, Δ$$2=0Thus$$ ∆ $$=$$ $$0$$

If $A, B, C, P, Q, R \in R$ and $Δ = |\begin{array}{lll} \cos (A + P) \cos (A + Q) \cos (A + R) \\ \cos (B + P) \cos (B + Q) \cos (B + R) \\ \cos (C + P) \cos (C + Q) \cos (C + R) \end{array}|$

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If A,B,C,P,Q,R∈R and Δ=cos⁡(A+P) cos⁡(A+Q) cos⁡(A+R)cos⁡(B+P) cos⁡(B+Q) cos⁡(B+R)cos⁡(C+P) cos⁡(C+Q) cos⁡(C+R)

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If $A, B, C, P, Q, R \in R$ and $Δ = |\begin{array}{lll} \cos (A + P) \cos (A + Q) \cos (A + R) \\ \cos (B + P) \cos (B + Q) \cos (B + R) \\ \cos (C + P) \cos (C + Q) \cos (C + R) \end{array}|$