If `tan theta+tan(60^(@)+theta)+tan(120^(@)+theta)=3,` Prove that:
`(3tantheta-tan^(3)theta)/(1-3tan^(2)theta)=1`

Explanation

`tantheta+tan(60^(@)+theta)+tan(120^(@)+theta)=3`
`rArr tantheta+tan(60^(@)+theta)-tan(60^(@)-theta)=3`
`tantheta+(tan60^(@)+tantheta)/(1-tan60^(@)tantheta)-(tan60^(@)-tantheta)/(1+tan60^(@)tantheta)=3`
`rArr tantheta+(sqrt(3)+tantheta)/(1-sqrt(3)tantheta)-(sqrt(3)-tantheta)/(1+sqrt(3)tantheta)=3`
`(tantheta(1-sqrt(3)tantheta)(1+sqrt(3)tantheta)+(sqrt(3)+tantheta)(1+sqrt(3)tantheta)-(sqrt(3)-tantheta)(1-sqrt(3)tantheta))/((1-sqrt(3)tantheta)(1+sqrt(3)tantheta))`
`rArr ((tantheta(1-3tan^(2)theta)+(sqrt(3)+3tantheta)+tantheta+sqrt(3)tan^(2)theta)-(sqrt(3)-3tantheta)-tantheta+sqrt(3)tan^(2)theta)/(1-3tan^(2)theta)=3`
`rArr (9tan0-3tan^(3)theta)/(1-3tan^(2)theta)=3`
`rArr (3 tantheta-tan^(3)theta)/(1-3tan^(2)theta)=1` Hence Proved.