Question

in ∆ABC PROVE THAT

bc cos²A/2 + ac cos²B/2 + ab cos²C/2​

Answer 1

$\huge{\underline{\tt{\red{SoluTion}}}}$

Formula

$\boxed{s= \dfrac{a+b+c}{2}} \\ \boxed{Cos\dfrac{A}{2} = \sqrt\dfrac{s(s-a)}{bc}}$

To prove

$bc Cos^2 \dfrac{A}{2} + ac Cos^2 \dfrac {B}{2} + ab Cos^2 \dfrac{C}{2} = S^2$

$\implies \cancel{bc} \bigg( \dfrac{s(s-a)}{\cancel{bc}}\bigg) + \cancel{ac} \bigg(\dfrac{s(s-b)}{\cancel{ac}}\bigg) + \cancel{ab} \bigg(\dfrac{s(s-c)}{\cancel{ab}} \bigg) \\ \\ \implies s (s-a+s-b+s-c) \\ \\ \implies s ( s +2s - a-b-c ) \\ \\ \implies s (s +2s -(a+b+c)) \\ \\ \implies s (s + 2s -2s) \\ \\ \huge\leadsto\boxed{s^2}$