In triangle ABC, ${cos}^{2} A + {cos}^{2} B - {cos}^{2} C = 1 - m sin A sin B sin C$ . Find $m$

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Solution

${cos}^{2} A + {cos}^{2} B - {cos}^{2} C = {cos}^{2} A + {sin}^{2} C - {sin}^{2} B$

$= {cos}^{2} A + sin (C + B) sin (C - B)$

$= 1 - {sin}^{2} A + sin A sin (C - B)$
$= 1 - sin A [sin A - sin (C - B)]$
$= 1 - sin A [sin (B + C) - sin (C - B)]$
$= 1 - 2 sin A sin B sin C$
Ans: m = 2

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