a^2( ^2B- ^2C)+b^2( ^2C- ^2A) +c^2( ^2A- ^2B)=0 - Vidyadip

Question

$\text{a}^2(\cos^2\text{B}-\cos^2\text{C})+\text{b}^2(\cos^2\text{C}-\cos^2\text{A})\\+\text{c}^2(\cos^2\text{A}-\cos^2\text{B})=0$

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Answer

$\text{a}^2(\cos^2\text{B}-\cos^2\text{C})+\text{b}^2(\cos^2\text{C}-\cos^2\text{A})\\+\text{c}^2(\cos^2\text{A}-\cos^2\text{B})=0$
$\text{LHS}=\text{a}^2(1-\sin^2\text{B}-1+\sin^2\text{C})\\\ \ \ \ \ \ \ \ \ \ \ +\text{b}^2(1-\sin^2\text{C}-1+\sin^2\text{A})\\ \ \ \ \ \ \ \ \ \ \ \ +\text{c}^2(1-\sin^2\text{A}-1+\sin^2\text{B})$
$=\text{a}^2(\sin^2\text{C}-\sin^2\text{B})+\text{b}^2(\sin^2\text{A}-\sin^2\text{C})\\+\text{c}^2(\sin^2\text{B}-\sin^2\text{A})$
$=\text{a}^2(\text{k}^2\text{c}^2-\text{k}^2\text{b}^2)+\text{b}^2(\text{k}^2\text{a}^2-\text{k}^2\text{c}^2)+\text{c}^2(\text{k}^2\text{b}^2-\text{k}^2\text{a}^2)$
$=\text{k}^2(\text{a}^2\text{c}^2-\text{a}^2\text{b}^2+\text{b}^2\text{a}^2-\text{b}^2\text{c}^2+\text{b}^2\text{c}^2-\text{a}^2\text{c}^2)$
$=\text{k}^2\times0 = 0=\text{RHS}$

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