Question:
Cos a + 2cos b +cos c =2 are in AP prove 2b =a + c
Answer:
Given, cos A + 2 cos B + cos C = 2
=> cos A + cos C = 2(1 - cos B)
=> 2 cos((A + C)/2) * cos((A-C)/2 = 4 sin2 (B/2)
=> 2 sin(B/2)cos((A-C)/2) = 4sin2 (B/2)
=> cos((A-C)/2) = 2sin (B/2)
=> cos((A-C)/2) =2cos((A+C)/2)
=> cos((A-C)/2) - cos((A+C)/2) = cos((A+C)/2)
=> 2sin(A/2)sin(C/2) = sin(B/2)
=> 2{√(s-b)(s-c)√bc} * {√(s-a)(s-b)√ab} = √(s-a)(s-c)√ac
=> 2(s - b) = b
=> a + b + c - 2b = b
=> a + c - b = b
=> a + c = 2b
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