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In any DeltaABC , find the value of (a...

In any `DeltaABC` , find the value of `(a^(2)sin(B-C))/(sinB+sinC)+(b^(2)sin(C-A))/(sinC+sinA)+(c^(2)sin(A-B))/(sinA+sinB)`

A

1

B

0

C

a+b+c

D

abc

Text Solution

Verified by Experts

The correct Answer is:
B
Let `(a)/(sinA)=(b)/(sinB)=(c)/(sinC)=k`
Then , `a=ksinA,b=ksinB,c=ksinC`
Now , `(a^(2)"sin"(B-C))/(sinB+sinC)+(b^(2)sin(C-A))/(sinC+sinA)+(c^(2)(A-B))/(sinA+sinB)`
`=(k^(2)sin^(2)Asin(B-C))/(sinB+sinC)+(k^(2)sin^(2)Bsin(C-A))/(sinC+sinA)`
`=k^(2){(sinAsin(B+C)sin(B-C))/(sinB+sinC)+(sinBsin(C+A)sin(C-A))/(sinC+sinA)+(sinCsin(A+B)sin(A-B))/(sinA+sinB)}`
`[becausesinA=sin(B+C),sinB+sin(C+A)andsinC=sin(A+B)]`
`=k^(2){(sinA(sin^(2)B-sin^(2)C))/(sinB+sinC)+(sinB(sin^(2)C-sin^(2)A))/(sinC+sinA)+(sinC(sin^(2)A-sin^(2)B))/(sinA+sinB)}`
`[becausesin(x+y)sin(x-y)=sin^(2)x-sin^(2)y]`
`=k^(2)[sinA(sinB-sinC)+sinB(sinC-sinA)+sinC(sinA-sinB)]`
`[because a^(2)-b^(2)=(a-b)(a+b)]`
`=k^(2)xx0=0`
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