I suspect you mean
1/8 sin(4t ) = 1/2 (cos³(t ) sin(t ) - sin³(t ) cos(t ))
On the right side, pull out a factor of cos(t ) sin(t ):
1/2 (cos³(t ) sin(t ) - sin³(t ) cos(t )) = 1/2 cos(t ) sin(t ) (cos²(t ) - sin²(t ))
Recall the double angle identities for sin and cos :
sin(2t ) = 2 sin(t ) cos(t )
cos(2t ) = cos²(t ) - sin²(t )
Then
… = 1/4 (2 cos(t ) sin(t )) (cos²(t ) - sin²(t ))
… = 1/4 sin(2t ) cos(2t )
… = 1/8 (2 sin(2t ) cos(2t ))
… = 1/8 sin(4t )
