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In DeltaABC if 2c cos^2(A/2)+ 2a Cos^2(C...

In `Delta`ABC if `2c cos^2(A/2)+ 2a Cos^2(C/2)` - 3b = 0. Prove that a, b, c are in AP

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Given equation is ,
`2c cos^2(A/2)+2a cos^2(C/2) - 3b = 0`
We know, `cos2x = 2cos^2x - 1 => 2cos^2x = 1+cos2x`
`:. c(1+cosA)+a(1+cosC) - 3b = 0`
`=>c(1+(b^2+c^2-a^2)/(2bc))+a(1+(a^2+b^2-c^2)/(2ab)) - 3b = 0`
`=>c((2bc+b^2+c^2-a^2)/(2bc))+a((2ab+a^2+b^2-c^2)/(2ab)) - 3b = 0`
`=>1/(2b)(2bc+b^2+c^2-a^2+2ab+a^2+b^2-c^2) - 3b =0`
`=>1/(2b)(2b^2+2b(c+a)) - 3b = 0`
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