In any triangle ABC prove the identities.
${sin}^{2} A + {sin}^{2} B - {sin}^{2} C = 2 sin A sin B cos C$ .

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Solution

L.H.S. $= {sin}^{2} A + sin (B + C) sin (B - C)$
$= s \in^{2} A + sin A sin (B - C)$
$= sin A [sin A + sin (B - C)]$
$= sin A [sin (B + C) + sin (B - C)]$
$= sin A [2 sin B cos C]$
$= 2 sin A sin B cos C$ .

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