1) Use double-angle identities to write the following expression, using trigonometric functions of x instead of 4x.
cos 4x
2) Use half-angle identities to write the following expression, using trigonometric functions of x instead of x/4.
sin x/4
cos4x = 2cos^2(2x)-1
= 2(2cos^2(x)-1)^2 - 1
just expand that
sin(x/4) = sqrt((1-cos(x/2))/2)
= sqrt((1-sqrt((1-cos(x))/2)/2
1) To write cos 4x in terms of trigonometric functions of x, we can use the double-angle identity for cosine, which states that:
cos(2θ) = 1 - 2sin²(θ)
We can rewrite 4x as 2 * (2x), so we have:
cos 4x = cos(2 * (2x))
Now, we can use the double-angle identity for cosine to express cos(2 * (2x)) in terms of trigonometric functions of 2x:
cos(2 * (2x)) = 1 - 2sin²(2x)
So, cos 4x can be written as 1 - 2sin²(2x) using trigonometric functions of x instead of 4x.
2) To write sin(x/4) in terms of trigonometric functions of x, we can use the half-angle identity for sine, which states that:
sin(θ/2) = ± √[(1 - cos(θ))/2]
In this case, we want to write sin(x/4) in terms of x. So, we can rewrite x/4 as (1/2) * (x/2), giving us:
sin(x/4) = sin((1/2) * (x/2))
Now, we can use the half-angle identity for sine to express sin((1/2) * (x/2)) in terms of trigonometric functions of x:
sin((1/2) * (x/2)) = ± √[(1 - cos(x/2))/2]
Therefore, sin(x/4) can be written as ± √[(1 - cos(x/2))/2] using trigonometric functions of x instead of x/4. Note that the ± sign represents the positive or negative solution, which depends on the quadrant in which x/4 lies.