Question

Sin3(2x) = 1/2sin(2x) [1 − cos(4x)]

Answer 1

I suppose we start with sin³(2*x) and we want to get:

(1/2)*sin(2*x)*(1 - cos(4*x))

Here we will use the relationships:

cos²(x) + sin²(x) = 1

and

cos²(x) = (1 + cos(2x))/2

Now let's start:

sin³(2*x) = sin(2*x)*sin²(2*x)

and we can write:

sin²(2*x) = 1 - cos²(2*x)

replacing that in the above equation we get:

sin³(2*x) = sin(2*x)*sin²(2*x) = sin(2*x)*(1 - cos²(2*x))

now we can use: cos²(2*x) = (1 + cos(2*2*x))/2

cos²(2*x) = (1 + cos(4*x))/2

if we replace that in the equation above we get:

sin³(2*x) = sin(2*x)*(1 - cos²(2*x)) = sin(2*x)*( 1 - (1 + cos(4*x))/2)

sin³(2*x) = sin(2*x)*( 1 - 1/2 - cos(4*x)/2)

sin³(2*x) = sin(2*x)*( 1/2 - cos(4*x)/2)

sin³(2*x) = sin(2*x)*( 1 - cos(4*x))/2

sin³(2*x) = (1/2)*sin(2*x)*( 1 - cos(4*x))