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In a triangle ABC, if sinAsin(B-C)=s9nCs...

In a triangle ABC, if `sinAsin(B-C)=s9nCsin(A-B),` then prove that `cotA ,cotB ,cotCa r einAdotPdot`

Text Solution

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`sin A sin(B-C)=sinC sin(A-B)`
`rArr (sin(B-C))/(sin C sinB)=(sin(A-B))/(sin A sin B)`
`rArr (sin B cos C-sin C cos B)/(sin C sinB)=(sin A cos B -sin B cos A)/(sinAsinB)`
`rArr cot C-cot B=cotB-cotA`
`rArr 2cot B=cot A+cot C`
`therefore cot A,cot B, cot C` are in AP.
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