4 (bc ^2 A 2 +ca ^2 B 2 +ab ^2 C 2 )=(a + b + c)^2 - Vidyadip

Question

$4\Big(\text{bc}\cos^2\frac{\text{A}}{2}+\text{ca}\cos^2\frac{\text{B}}{2}+\text{ab}\cos^2\frac{\text{C}}{2}\Big)=(\text{a + b + c})^2$

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Answer

$4\Big(\text{bc}\cos^2\frac{\text{A}}{2}+\text{ca}\cos^2\frac{\text{B}}{2}+\text{ab}\cos^2\frac{\text{C}}{2}\Big)=(\text{a + b + c})^2$$\text{LHS}=4\Big(\text{bc}\cos^2\frac{\text{A}}{2}+\text{ca}\cos^2\frac{\text{B}}{2}+\text{ab}\cos^2\frac{\text{C}}{2}\Big)$
$=2\Big(\text{bc.}2\cos^2\frac{\text{A}}{2}+\text{ca.}2\cos^2\frac{\text{B}}{2}+\text{ab}.2\cos^2\frac{\text{C}}{2}\Big)$
$=2(\text{bc.}(1-\cos\text{A})+\text{ca.}(1-\cos\text{B})+\text{ab.}(1-\cos\text{C}))$
$=2\text{bc}-2\text{bc}\cos\text{A}+2\text{ca}-2\text{ca}\cos\text{B}+2\text{ab}-2\text{ab}\cos\text{C}$
$=2\text{bc}-2\text{bc}\frac{\text{b}^2+\text{c}^2-\text{a}^2}{2\text{bc}}+2\text{ca}-2\text{ca}\frac{\text{a}^2+\text{c}^2-\text{b}^2}{2\text{ca}}+2\text{ab}$
$-2\text{ab}\frac{\text{b}^2+\text{a}^2-\text{c}^2}{2\text{ab}}\text{[cos rule]}$
$=2\text{bc}-\text{b}^2-\text{c}^2+\text{a}^2+2\text{ca}-\text{a}^2-\text{c}^2+\text{b}^2+2\text{ab}-\text{b}^2-\text{a}^2+\text{c}^2$
$=\text{a}^2+\text{b}^2+\text{c}^2+2\text{ab}+2\text{ab}+2\text{ca}$
$=(\text{a + b + c})^2=\text{RHS}$

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