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If tan^3A+tan^3B+tan^3C=3tanA tanB tanC ...

If `tan^3A+tan^3B+tan^3C=3tanA tanB tanC ,` then prove that triangle ABC is an equilateral triangle.

Text Solution

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`tan^(3)A+tan^(3)B+tan^(3)C=3 tan A.tan B.tanC`
`rArr (tan A+tan B+tan C)((tan A-tan B^(2))`
`+(tan B-tanC)^(@)+(tan C-tanA^(2))=0`
In `triangle ABC`
`tan A+tan B+tanC=tan A and B tan C`
So, `tan A+tan B+tan Cne0`
`therefore tan A=tan B=tan C`
`therefore tan A=tan B=tan C`
`rArr A-B=C=60^(@)`
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