a^2=(b + c)^2-4bc ^2 A 2 - Vidyadip

Question

$\text{a}^2=(\text{b + c})^2-4\text{bc}\cos^2\frac{\text{A}}{2}$

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Answer

$\text{RHS}=(\text{b + c})^2-4\text{bc}\cos^2\frac{\text{A}}{2}$
$=\text{b}^2+\text{c}^2+2\text{bc}-4\text{bc}\Big(\frac{1+\cos\text{A}}{2}\Big)$
$=\text{b}^2+\text{c}^2+2\text{bc}-2\text{bc}(1+\cos\text{A})$
$=\text{b}^2+\text{c}^2+2\text{bc}(1-1-\cos\text{A})$
$=\text{b}^2+\text{c}^2-2\text{bc}\cos\text{A}$
$=\text{b}^2+\text{c}^2-2\text{bc}\Big(\frac{\text{b}^2+\text{c}^2-\text{a}^2}{2\text{bc}}\Big)$ $\Big(\because\cos\text{A}=\frac{\text{b}^2+\text{c}^2-\text{a}^2}{2\text{bc}}\Big)$
$=\text{b}^2+\text{c}^2-\text{b}^2-\text{c}^2+\text{a}^2$
$=\text{a}^2=\text{LHS}$
Hence proved.

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