Determinanats...?

Question

evaluate the determinant ----


| 1 cos(B-A) cos(C-A) |
| cos(A-B) 1 cos(A-C) |
| cos(A-C) cos(B-C) 1 |

order - 3 x 3

for further undestanding ---
elements of R1 are --- 1 , cos(B-A) , cos(C-A)
elements of R2 are --- cos(A-B) , 1 , cos(A-C)
elements of R3 are --- cos(A-C) , cos(B-C) , 1


pls. solve it sooon .... thnx

Answer 1

| 1 cos(B-A) cos(C-A) |
| cos(A-B) 1 cos(A-C) |
| cos(A-C) cos(B-C) 1 |
∆ = 1 + cos(A - B)cos^2(A - C) + cos(A - B)cos(B - C)cos(A - C) - cos^2(A - C) - cos^2(A - B) - cos(A - C)cos(B - C)
cos (s – t) = cosscost + sinssint
1 + (cosAcosB + sinAsinBt) (cosAcosC + sinAsinC)^2 + (cosAcosB + sinAsinB)(cosBcosC + sinBsinC)(cosAcosC + sinAsinC) - (cosAcosC + sinAsinC)^2 - (cosAcosB + sinAsinB)^2 - (cosAcosB + sinAsinB)(cosBcosC + sinBsinC) =

1 + (cosAcosB + sinAsinBt) ((cosAcosCcosAcosC + cosAcosCsinAsinC) + (sinAsinCcosAcosC + sinAsinCsinAsinC) + (cosAcosB + sinAsinB)((cosBcosCcosAcosC + cosBcosCsinAsinC) + (sinBsinCcosAcosC + sinBsinCsinAsinC) - ((cosAcosCcosAcosC + cosAcosCsinAsinC) + sinAsinC(sinAsinCcosAcosC + sinAsinCsinAsinC) - (cosAcosB(cosAcosBcosAcosB + cosAcosBsinAsinB) + (sinAsinBcosAcosB + sinAsinBsinAsinB) - (cosAcosBcosBcosC + cosAcosBsinBsinC) + (sinAsinBcosBcosC + sinAsinBsinBsinC) =

1 + (cosAcosB + sinAsinBt) (cosAcosC(cosAcosC + sinAsinC) + sinAsinC(cosAcosC + sinAsinC) + (cosAcosB + sinAsinB)(cosBcosC(cosAcosC + sinAsinC) + sinBsinC(cosAcosC + sinAsinC) - (cosAcosC(cosAcosC + sinAsinC) + sinAsinC(cosAcosC + sinAsinC) - (cosAcosB(cosAcosB + sinAsinB) + sinAsinB(cosAcosB + sinAsinB) - cosAcosB(cosBcosC + sinBsinC) + sinAsinB(cosBcosC + sinBsinC) =

1 + (cosAcosB + sinAsinBt) ((cosAcosCcosAcosC + cosAcosCsinAsinC) + (sinAsinCcosAcosC + sinAsinCsinAsinC) + (cosAcosB + sinAsinB)((cosBcosCcosAcosC + cosBcosCsinAsinC) + (sinBsinCcosAcosC + sinBsinCsinAsinC) - ((cosAcosCcosAcosC + cosAcosCsinAsinC) - (sinAsinCcosAcosC + sinAsinCsinAsinC) - ((cosAcosBcosAcosB + cosAcosBsinAsinB) - (sinAsinBcosAcosB + sinAsinBsinAsinB) - (cosAcosBcosBcosC + cosAcosBsinBsinC) - (sinAsinBcosBcosC + sinAsinBsinBsinC) =

1 + ((cosAcosCcosAcosC(cosAcosB + sinAsinBt) + 2cosAcosCsinAsinC(cosAcosB + sinAsinBt) + sinAsinCsinAsinC(cosAcosB + sinAsinBt)) ++ ((cosBcosCcosAcosC(cosAcosB + sinAsinB) + cosBcosCsinAsinC(cosAcosB + sinAsinB)) + (sinBsinCcosAcosC(cosAcosB + sinAsinB) + sinBsinCsinAsinC(cosAcosB + sinAsinB)) - ((cosAcosCcosAcosC + 2cosAcosCsinAsinC + sinAsinCsinAsinC) - ((cosAcosBcosAcosB + 2cosAcosBsinAsinB + sinAsinBsinAsinB) - (cosAcosBcosBcosC + cosAcosBsinBsinC) - (sinAsinBcosBcosC + sinAsinBsinBsinC) =

1 + cosAcosCcosAcosCcosAcosB + cosAcosCcosAcosCsinAsinB + 2cosAcosCsinAsinCcosAcosB + 2cosAcosCsinAsinCsinAsinB + sinAsinCsinAsinCcosAcosB + sinAsinCsinAsinCsinAsinB + cosBcosCcosAcosCcosAcosB + cosBcosCcosAcosCsinAsinB + cosBcosCsinAsinCcosAcosB + cosBcosCsinAsinCsinAsinB + sinBsinCcosAcosCcosAcosB + sinBsinCcosAcosCsinAsinB + sinBsinCsinAsinCcosAcosB + sinBsinCsinAsinCsinAsinB - cosAcosCcosAcosC - 2cosAcosCsinAsinC - sinAsinCsinAsinC - cosAcosBcosAcosB - 2cosAcosBsinAsinB - sinAsinBsinAsinB - cosAcosBcosBcosC - cosAcosBsinBsinC - sinAsinBcosBcosC - sinAsinBsinBsinC) =

1 + cos^2Acos^2CcosAcosB + cos^2Acos^2CsinAsinB + 2cos^2AcosCsinAsinCcosB + 2cosAcosCsin^2AsinCsinB + sin^2Asin^2CcosAcosB + sin^2Asin^2CsinAsinB + cos^2Bcos^2Ccos^2A + cos^2CcosAcosBsinAsinB + sinAsinCcosAcos^2BcosC + cosBcosCsin^2AsinBsinC + cos^2AcosBcosCsinBsinC + cosAcosCsinAsin^2BsinC + cosAcosBsinAsinBsin^2C + sin^2Asin^2Bsin^2C - cos^2Acos^2C - 2cosAcosCsinAsinC - sin^2Asin^2C - cos^2Acos^2B - 2cosAcosBsinAsinB - sin^2Asin^2B - cosAcos^2BcosC - cosAcosBsinBsinC - cosBcosCsinAsinB - sinAsin^2BsinC) =

Answer 2

the determinant is 0 because there is no value given for C, B & A....so you may refer the cos of everything = 1...