In a ` DeltaA B C ,` prove the following : `4\ (b c cos^2(A/2)+c acos^2(B/2)+a bcos^2(C/2)\ )\ =(a+b+c)^2`

Explanation

LHS

`=4(b c cos ^{2} frac{A}{2}+c a cos ^{2} frac{B}{2}+a b cos ^{2} frac{C}{2}) `

`=4[b c(frac{1+cos A}{2})+c a(frac{1+cos B}{2})+a b(frac{1+cos C}{2})] `

=2 b c+2 b c cos A+2 c a+2 c a cos B+2 a b+2 a b cos C

`=2(a b+b c+c a)+2 b c(frac{b^{2}+c^{2}-a^{2}}{2 b c})+2 c a(frac{c^{2}+a^{2}-b^{2}}{2 c a})+2 a b(frac{a^{2}+b^{2}-c^{2}}{2 a b})`

`=2(a b+b c+a c)+b^{2}+c^{2}-a^{2}+c^{2}+a^{2}-b^{2}+a^{2}+b^{2}-c^{2}`

`=a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 a c`

`=(a+b+c)^{2}=text { RHS }`

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