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In DeltaABC, prove that: asin(B-C)+bsi...

In `DeltaABC`, prove that:
`asin(B-C)+bsin(C-A)+csinA-B)=0`

Text Solution

Verified by Experts

1st term =`asin(B-C)`
`=ksinA.sin(B-C)`
`=ksin{180^(@)-(B+C)}sin(B-C)`
`ksin(B+C).sin(B-C)`
`=ksin^(2)B-sin^(2)C`
Similarly,
2nd term `=k(sin^(2)C-sin^(2)A)`
3rd term `=k(sin^(2)A-sin^(2)B)`
Now L.H.S. `=asin(B-C)+bsin(C-a)+csin(A-B)`
`=k(sin^(2)B-sin^(2)C)+k(sin^(2)C-sin^(2)A)+k(sin^(2)A-sin^(2)B)`
`ksin^(2)B-ksin^(2)C-ksin^(2)A+ksin^(2)A-ksin^(2)B` =0 = RHS Hence Proved.
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