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Question

In ABC with usual notations, acosA+bcosB+ccosCa+b+c=12, then the value of sinA+sinB+sinCsinAsinBsinC is

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Solution

acosA+bcosB+ccosCa+b+c=12
acosA+bcosB+ccosC=a+b+c2
Using sine rule, we get
asinA=bsinB=csinC=k (say)

Now,
2[sinAcosA+sinBcosB+sinCcosC]=sinA+sinB+sinCsin2A+sin2B+sin2C=sinA+sinB+sinC
In a ABC,
sin2A+sin2B+sin2C=4sinAsinBsinC
So, 4sinAsinBsinC=sinA+sinB+sinC
sinA+sinB+sinCsinAsinBsinC=4

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