How do you divide 85 by 41 with remainder

First, find how many times can fit into .

is greater than , so we need to go back down to .

Now, subtract to get a remainder of .

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 )