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Prove that: t a nA+tan(60^0+A)-t a n(60^...

Prove that: `t a nA+tan(60^0+A)-t a n(60^0-A)=3t a n3A`

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LHS `=tanA=tan(60^(@)+A)+tan(120^(@)+A)`
`=tanA+tan(60^(@)+A)+tan(180^(@)-(60^(@)-A)]`
`=tan A+tan(60^(@)+A)-tan(60^(@)-A) [because tan(180^(@)-theta) =-tan theta]`
`=tan A+(tan60^(@)+tanA)/(1-tan60^(@)tanA)-(tan60^(@)-tanA)/(1+tan60^(@)tanA)=tan A+(sqrt(3)+tanA)/(1-sqrt(3)tanA)(1+sqrt(3)tanA))`
`=tanA+(8tanA)/(1-3tan^(2)A)=(tanA-3tan^(2)A+8tanA)/(1-3tan^(2)A)`
`=(9tanA-3tan^(3)A)/(1-3tan^(2)A)=3((3tanA-tan^(3)A)/(1-3tan^(2)A))=3 tan 3A=RHS`.
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