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Question

sin(B+C−A)+sin(C+A−B)+sin(A+B−C)=

A
4 sinAsinBsinC
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B
4 cosAcosBcosC
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C
4 sin(A/2)sin(B/2)sin(C/2)
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D
None of these
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Solution

The correct option is A 4 sinAsinBsinC
According to question,
sin(B+CA)+sin(C+AB)+sin(A+BC)weknow,,,ΔA+B+C=1800=sin(πAA)+sin(πBB)+sin(πCC)=sin(π2A)+sin(π2B)+sin(π2C)=sin2A+sin2B+sin2C=2sin(A+B)cos(AB)+sin2C=2sin(πC)cos(AB)+sin2C=2sinCcos(AB)+sin2C=2sinCcos(AB)+2sinCcosC=2sinC[cos(AB)+cosC]=2sinC[(AB)+cos(π(A+B))]=2sinC[cos(AB)+(1)cos(A+B)]=2sinC[COS(AB)cos(A=B)]=2sinC(1)2sin(AB+A+B)2×sin(ABAB)2=2sinC2sinAsin(B)[weknow,sin(θ)=sinθ]=4sinC.sinA.sinB4sinAsinBsinC
So,that the correct option is A.

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