Answer :

RHS = sin2A + sin2(A -B) – 2 sinA cosB sin(A -B)


= sin2A + sin(A -B) [sin(A –B) – 2 sinA cosB]


We know that sin(A –B) = sinA cosB – cosA sinB


= sin2A + sin(A -B) [sinA cosB – cosA sinB – 2 sinA cosB]


= sin2A + sin(A -B) [-sinA cosB – cosA sinB]


= sin2A - sin(A -B) [sinA cosB + cosA sinB]


We know that sin(A +B) = sinA cosB + cosA sinB


= sin2A – sin(A –B) sin(A +B)


= sin2A – sin2A + sin2B


= sin2B = LHS


Hence proved.


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