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Question

Assertion (A): lnΔABC,∑cosAsinBsinC=2.Reason(R): lnΔABC,sinA+sinB+sinC=4cosA2cosB2cosC2.

A
A is true, R is true and R is correct explanation of A
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B
A is true, R is true and R is not correct explanation of A
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C
A is true, R is false
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D
A is false, R is true.
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Solution

The correct option is B A is true, R is true and R is not correct explanation of AcosAsinBsinC+cosBsinAsinC+cosCsinAsinB=2sinAcosA+sinBcosB+sinCcosCsinAsinBsinC=2sin2A+sin2B+sin2C=4sinAsinBsinC2sin(A+B)cos(A−B)+sin2C=4sinAsinBsinC2sin(A+B)cos(A−B)+sin2C=4sinAsinBsinCcos(A−B)+cosC=2sinAsinBcos(A−B)−cos(A+B)=2sinAsinBHence, proved.2sin(A+B2)cos(A−B2)+2sinC2cosC22cosC2[cos(A−B2)+cos(A+B2)]=4cosA2cosB2cosC2 hence proved.Therefore, A is true, R is true and R is not correc explanation of A.

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