Answer :

Let a, b, c be the sides of any triangle ABC. Then by applying the sine rule, we get




c = k sin C


Similarly, b = k sin B


And a = k sin A


Here we will consider LHS, so we get


LHS = b sin B – c sin C


Substituting corresponding values in the above equation, we get


= k sin B sin B –k sin C sin C


= k (sin2 B – sin2 C )……….(ii)


But,



Substituting the above values in equation (ii), we get


= k(sin(B + C) sin(B - C))


But A + B + C = π B + C = π –A, so the above equation becomes,


= k(sin(π –A) sin(B - C))


But sin (π - θ) = sin θ


= k(sin(A) sin(B - C))


From sine rule, a = k sin A, so the above equation becomes,


= a sin(B - C) = RHS


Hence proved


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