Answer :

Need to prove: a sin A – b sin B = c sin (A – B)


Left hand side,


a sin A – b sin B


(b cosC + c cosB) sinA – (c cosA + a cosC) sinB


b cosC sinA + c cosB sinA – c cosA sinB – a cosC sinB


c(sinA cosB – cosA sinB) + cosC(b sinA – a sinB)


We know that, where R is the circumradius.


Therefore,


c(sinA cosB – cosA sinB) + cosC(2R sinB sinA – 2R sinA sinB)


c(si nA cosB – cosA sinB)


c sin (A – B)


Right hand side. [Proved]


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