साबित करें कि
`tanalpha.tan ( 60^(@)-alpha)tan ( 60^(@) + alpha) = tan 3 alpha = tan 3 alpha `

Explanation

L.H.S. `= ( sin alpha)/(cos alpha) . ( sin ( 60^(@) - alpha))/(cos ( 60^(@) - alpha)) . ( sin ( 60^(@) + alpha))/(cos ( 60^(@) + alpha))`
`= ( sin alpha( sin^(2) ( 60^(@) - sin^(2) alpha)))/(cos alpha( cos^(2) 60^(@) - sin^(2) alpha))= ( sinalpha((3)/(4) - sin^(2) alpha))/(cos alpha ((1)/(4)- sin^(2)alpha))`
`[ :' sin ( A+B )sin ( A-B) = sin^(2) A - sin^(2) B ` तथा `cos ( A+B) cos ( A-B) = cos^(2) A - sin^(2) B ]`
`= (sinalpha( 3-4sin^(2) alpha))/(cos alpha ( 1-4 sin^(2) alpha)) = ( 3 sinalpha-4sin^(3) alpha)/(cos alpha [ 1-4( 1- cos^(2) alpha)])`
`= ( 3 sinalpha - 4 sin^(3) alpha)/(cos alpha ( 4 sin^(2) alpha - 3 ))= ( 3 sinalpha- 4 sin^(3) alpha)/( 4 cos ^(3) alpha - 3 cos alpha)`
`= ( sin 3 alpha)/( cos 3 alpha ) = tan 3 alpha`