In triangle $A B C,$ ${cos}^{2} A + {cos}^{2} B - {cos}^{2} C = 1 - m sin A sin B cos C$ . The value of $m$ is

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Solution

$= {cos}^{2} A + {cos}^{2} B - {cos}^{2} C$
$= {cos}^{2} A + (1 - {sin}^{2} B) - (1 - {cos}^{2} C)$
$= {cos}^{2} A + {sin}^{2} C - {sin}^{2} B$
$= {cos}^{2} A + sin (C + B) sin (C - B)$
$= 1 - {sin}^{2} A + sin (π - A) sin (C - B)$
$= 1 - {sin}^{2} A + sin A sin (C - B)$
$= 1 - sin A [sin A - sin (C - B)]$
$= 1 - sin A [sin (B + C) - sin (C - B)]$
$= 1 - 2 sin A cos \frac{2 C}{2} sin \frac{2 B}{2}$
$= 1 - 2 sin A sin B cos C$
On comparing we get,
$m = 2$

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