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4 Jun 2019

Prove that, CosA + CosB + CosC + Cos(A + B + C) = 4 Cos(A + B)/2 Cos(B +C)/2 Cos(C+A)/2

Prove that,   CosA + CosB + CosC + Cos(A + B + C) =  4 Cos(A + B)/2 Cos(B +C)/2 Cos(C+A)/2


Sol. 

  CosA + CosB + CosC + Cos(A + B + C)

=  2Cos(A+B)/2 Cos(A-B)/2   +2cos(C+A+B+C)/2 Cos(C-A-B-C)/2

= 2Cos(A+B)/2 Cos(A-B)/2   +2cos(A+B+2C)/2 Cos(-A-B)/2 

= 2Cos(A+B)/2 Cos(A-B)/2     +2cos(A+B+2C)/2 Cos(A+B)/2

= 2Cos(A+B)/2[Cos(A-B)/2       +Cos(A+B+2c)/2]

= 2Cos(A+B)/2[2Cos(2A+2C)/4               Cos(-2B-2C)/4]

= 2Cos(A+B)/2[2Cos(A+C)/2 Cos(B+C)/2]

= 4Cos(A+B)/2 Cos(B+C)/2 Cos(C+A)/2

Hence proved.

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