साबित करें कि
`sin^(3) alpha+ sin^(3) ((2pi)/(3)+alpha) + sin^(3) ((4pi)/(3)+alpha) = - (3)/(4)sin3alpha `

Explanation

`:. sin^(3) theta = (1)/(4)(3 sintheta- sin 3 theta)`
`:.` L.H.S. `=(1)/(4)[3sin alpha - sin 3 alpha ] + (1)/(4) [ 3 sin ((2pi)/(3)+alpha) - sin ( 2 pi + 3 alpha)] + (1)/(4) [ 3 sin ((4pi)/(3)+ alpha) - sin ( 4 pi + 3 alpha)]`
`= (1)/(4) [ 3 sin alpha - sin 3 alpha] + (1)/(4) [ 3 sin ((2pi)/(3) +alpha) - sin 3 alpha] + (1)/(4) [ 3 sin (( 4pi)/(3) + alpha) - sin 3 alpha]`
`[ :' sin ( 2pi + theta) = sin theta ` and ` sin ( 4pi + theta) = sin theta ]`
`= (3)/(4) sin alpha + (3)/(4) [ sin ((2pi)/(3) + alpha) + sin ((4pi)/(3)+alpha)]- (1)/(4) ( sin 3 alpha+ sin3alpha+ sin 3 alpha)`
`= (3)/(4) sin alpha + (3)/(4) 2sin ( pi+alpha) cos . ( pi)/(3) - (3)/(4) sin 3alpha`
[सूत्र `sinC + sin D ` के प्रयोग से ]
`= (3)/(4) sin alpha + (3)/(4) . 2. ( - sin alpha ) . (1)/(2) - (3)/(4) sin 3 alpha `
` = (3)/(4) sin alpha - (3)/(4) sin alpha - (3)/(4) sin 3 alpha = - (3)/(4) sin 3 alpha `