Prove $sin (A + B + C) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C - sin A sin B sin C$

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Solution

Formulae:
$sin (B + C) = sin B cos C + cos B sin C$
$cos (B + C) = cos B cos C - sin B sin C$

$\Rightarrow$ $sin (A + B + C) =$ $s i n A cos (B + C) + cos A sin (B + C)$

$\Rightarrow$ $sin A cos B cos C - sin A sin B sin C + cos A sin B cos C + cos A cos B sin C$

$\Rightarrow s i n A cos B cos C + cos A sin B cos C + cos A cos B sin C - sin A sin B sin C$

Hence proved

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