Math, asked by manasetu2002, 11 months ago

if the side lengths a,b,c are in A.P. then prove that cos(A-C)/2 = 2sin B/2​

Answers

Answered by amitnrw
6

Answer:

cos(A-C)/2 = 2sin B/2​

Step-by-step explanation:

a/SinA = b/SinB = c/SinC = k

a = kSinA

b =kSinB

c = kSinC

a,b,c are in A.P

=> 2kSinB = kSinA + k SinC

=> 2SinB = SinA + SinC

using Sin2x = 2SinxCosx

& Sinx + Siny  = 2Sin((x + y)/2)Cos((x - y)/2)

=> 2 (2Sin(B/2)Cos(B/2)) = 2 Sin((A + C)/2) Cos((A-C)/2)

A+ B + C = 180° = π  => A + C = π- B   => (A + C)/2 = π/2 - B/2

=> Sin( (A + C)/2) = Sin(π/2 - B/2)= Cos (B/2)

=> 2Sin(B/2)Cos(B/2) =  Cos (B/2) Cos((A-C)/2)

cancelling Cos(B/2) from both sides

=>  2Sin(B/2) =  Cos((A-C)/2)

=> cos(A-C)/2 = 2sin B/2​

QED

Proved

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