यदि ` tan theta + tan (60^(@) + theta ) + tan ( 120^(@)+ theta )=3`,
तो सिद्ध कीजिए कि - ` (3 tan theta - tan ^(3) theta )/( 1-3 tan^(2) theta )=1`

Explanation

` tan theta + tan ( 60^(@)+ theta) + tan ( 20^(@)+theta) = 3`
` rArr tan theta + tan ( 60^(@) + theta) + tan {180^(@) - (60^(@) - theta)} = 3`
` rArr tan theta + tan ( 60 ^(@) + theta) - tan (60^(@) - theta) = 3`
` rArr tan theta + (tan 60^(@) + tan theta)/(1-tan 60^(@) tan theta)- (tan 60^(@) - tan theta)/(1+ tan 60^(@) tan theta)=3`
` rArr tan theta + (sqrt(3)+tan theta)/(1-sqrt(3)tan theta) - (sqrt(3)- tan theta)/(1+sqrt(3)tantheta)=3`
` tan theta ( 1- sqrt(3) tan theta) ( 1+ sqrt(3) tan theta)`
` + (sqrt(3) + tan theta) ( 1 + sqrt(3) tan theta)`
` rArr (-(sqrt(3)-tan theta)(1-sqrt(3)tan theta))/ ((1-sqrt(3)tan theta)( 1 = sqrt(3) tan theta) )=3`
` tan theta ( 1- 3 tan ^(2) theta) + ( sqrt(3) + 3 tan theta`
` + tan theta + sqrt(3) tan^(2) theta) - (sqrt(3)- 3 tan theta`
` rArr (-tan theta + sqrt(3) tan^(2) theta)/(1-3 tan^(2)theta)=3`
` rArr (3 tan theta - tan^(2) theta)/(1-3tan^(2) theta)=1`