1.

Prove that: `tanA + tan(60^(@)+A) + tan(120^(@)+A)=3tan3A`

Answer» LHS `=tanA+tan(60^(@)+A)+tan(120^(@)+A)`
`=tanA+tan(60^(@)+A)-tan(60^(@)-A)` `(therefore tan(180^(@)-theta) = -tantheta)`
`=tanA+(tan60^(@)+tanA)/(1+tan60^(@)tanA) = tanA+(sqrt(3)+tanA)/(1-sqrt(3)tanA)-(sqrt(3)-tanA)/(1+sqrt(3)tanA)`
`=tanA+(sqrt(3)+tanA+3tanA+sqrt(3)tan^(2)A-sqrt(3)+tanA+3tanA-sqrt(3)tan^(2)A)/((1-sqrt(3)tanA)(1+sqrt(3)tanA)`
`=tan+(8 tanA)/(1-3tan^(2)A) = (tanA-3tan^(3)A+8tanA)/(1-3tan^(2)A)`
`=(9tanA-3tan^(3)A)/(1-3tan^(2)A) = (tanA-3tan^(3)A+8tanA)/(1-3tan^(2)A)`
`=(9tanA-3tan^(3)A)/(1-3tan^(2)A) = 3(3tanA-tan^(3)A)/(1-3tan^(2)A) = 3tan3A`=RHS


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