Question

Cos(a-b)+cos(b-c)+cos(c-a)

Answer 1

Combine the first two terms   and expand the last term  using trigonometric identities.

 

2 Cos (a-c)/2 cos (a-2b+c)/2  + 2 cos^2 (a –c)/2  - 1

2 cos (a-c)/2  [ cos (a-2b+c)/2 + cos (a-c)/2 ] – 1

4 cos (a-c)/2   cos (a-b)  cos (c-b)  - 1

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Another simplification:

 

a-b + b –c  + c - a  = 0   =>  a-c = (a-b)  +  (b –c)

=> Cos(a-b)+ cos(b-c) +  [ cos (a-b) cos(b-c) – sin (a-b) sin (b-c) ]

=> Cos(a-b) (1 + cos(b-c) ) + cos (b-c) – sin(a-b) sin(b-c)

=>   2 Cos(a-b) cos^2 (b-c)/2   + 2 cos^2  (b-c)/2  - sin(a-b) sin(b-c)  -1

=>  4 cos^2 (b-c)/2   cos^2 (a-b)/2    - 4 cos(a-b)/2 cos(b-c)/2  sin(a-b/2) sin(b-c)/2  -1  

=>  4 cos (b-c)/2  cos(a-b)/2 [cos (b-c)/2  cos(a-b)/2 - sin(a-b/2) sin(b-c)/2  ] – 1

=>  4 cos (b-c)/2  cos(a-b)/2 cos(a-c)/2 – 1