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

cos(4x) = 2cos^2(2x)-1

cos2x = 2cos^2(x) - 1

cos(4x) = 2(2cos^2(x)-1)^2 - 1
= 2(4cos^4(x) - 4cos^2(x)+1) - 1
= 8cos^4(x) - 8cos^2(x) + 1
or,
sin^4(x) + cos^4(x) -6 sin^2(x)cos^2(x)
or a bunch of other expressions

cos(x/2) = sqrt[(1 + cos(x))/2
sin(x/4) = sqrt[(1 - cos(x/2))/2)
= sqrt([1-(sqrt[(1+cos(x))/2)]/2)

1) To express cos 4x using trigonometric functions of x, we can use the double-angle identity for cosine:

cos(2θ) = 2cos^2(θ) - 1

Let's set θ = 2x. Then, we have:

cos 4x = cos(2(2x)) = 2cos^2(2x) - 1

Now, we can express cos^2(2x) using another double-angle identity:

cos^2(2x) = (1 + cos(4x))/2

Substituting this into the previous equation, we get:

cos 4x = 2((1 + cos(4x))/2) - 1

Simplifying further:

cos 4x = 1 + cos(4x) - 1

Which further simplifies to:

cos 4x = cos 4x

Therefore, we do not have a simplified expression for cos 4x in terms of trigonometric functions of x.

2) To express sin(x/4) using trigonometric functions of x, we can use the half-angle identity for sine:

sin(θ/2) = ±√((1 - cosθ) / 2)

Let's set θ = x/2. Then, we have:

sin(x/4) = sin((x/2)/2) = ±√((1 - cos(x/2)) / 2)

Since x/2 is positive, we can take the positive square root:

sin(x/4) = √((1 - cos(x/2)) / 2)

Therefore, the expression sin(x/4) can be written in terms of trigonometric functions of x as √((1 - cos(x/2)) / 2).

1) To write cos 4x using trigonometric functions of x, we can use the double-angle identity for cosine.

The double-angle identity for cosine is:
cos(2θ) = cos²(θ) - sin²(θ)

Let's substitute 4x as 2θ in this identity:
cos 4x = cos²(2x) - sin²(2x)

Now, let's express sin²(2x) and cos²(2x) using trigonometric functions of x using the double-angle identities for sine and cosine.

The double-angle identity for sine is:
sin(2θ) = 2sin(θ)cos(θ)

Let's substitute 2x as 2θ in this identity:
sin(2x) = 2sin(x)cos(x)

Now, let's express sin²(2x) using trigonometric functions of x:
sin²(2x) = (2sin(x)cos(x))²
= 4sin²(x)cos²(x)

The double-angle identity for cosine is:
cos(2θ) = cos²(θ) - sin²(θ)

Let's substitute 2x as 2θ in this identity:
cos(2x) = cos²(x) - sin²(x)

Now, let's express cos²(2x) using trigonometric functions of x:
cos²(2x) = (cos²(x) - sin²(x))²
= cos⁴(x) - 2cos²(x)sin²(x) + sin⁴(x)

Finally, we can substitute sin²(2x) and cos²(2x) in our original expression cos 4x:
cos 4x = cos²(2x) - sin²(2x)
= cos⁴(x) - 2cos²(x)sin²(x) + sin⁴(x) - 4sin²(x)cos²(x)

So, cos 4x using trigonometric functions of x is cos⁴(x) - 2cos²(x)sin²(x) + sin⁴(x) - 4sin²(x)cos²(x).

2) To write sin x/4 using trigonometric functions of x, we can use the half-angle identity for sine.

The half-angle identity for sine is:
sin(θ/2) = ±√((1 - cos(θ)) / 2)

Let's substitute x/4 as θ/2 in this identity:
sin(x/4) = ±√((1 - cos(x)) / 2)

Since we are using x instead of θ, we can write the expression as:
sin x/4 = ±√((1 - cos(x)) / 2)

So, sin x/4 using trigonometric functions of x is ±√((1 - cos(x)) / 2).