In any `DeltaABC`, prove that `((a-b))/c"cos"C/2="sin"((A-B))/2`

1.	In any `DeltaABC`, prove that `((a-b))/c"cos"C/2="sin"((A-B))/2`
Answer» By the sine rule, we have `a/("sin A")=b/("sin B")=c/("sin C")="k(sya)"` `rArr" a=k sin A, b = k sin B and c = k sin C".` `:." LHS"=((a-b))/c"cos"C/2` `=(("k sin A-k sin B"))/(" sin C")"cos"C/2=("k(sin A-sinB)")/("k sin C")"cos"C/2` `=("sin A - sin B")/("sin C")"cos"C/2=("2cos"((A+B))/2"sin"((A-B))/2)/("2sin"C/2"cos"C/2)"cos"C/2" "[because("sin C- sin D")="2cos"((C+D))/2"sin"((C-D))/2]` `=("cos"(pi/2-C/2)"sin"((A-B))/2)/("sin"C/2)" "[because(A+B)/2=(pi/2-C/2)]` `=("sin"C/2"sin"((A-B))/2)/("sin"C/2)="sin"((A-C))/2="RHS."` `"Hence, "((a-b))/c"cos"C/2="sin"((A-B))/2.`

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