2(bc +ca +ab )=a^2+b^2+c^2 - Vidyadip

Question

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

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Answer

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

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