Rewrite in terms of an expression containing only cosines to the power 1.

4 cos^6x

3(14+27)

To rewrite 4 cos^6x in terms of an expression containing only cosines to the power 1, we can use the trigonometric identity:

cos^2x = (1/2) (1 + cos2x)

Let's apply this identity multiple times to simplify the expression:

4 cos^6x = 4 (cos^2x)^3

Using the identity, we can rewrite cos^2x as:

cos^2x = (1/2) (1 + cos2x)

Substituting this back into the equation, we have:

4 (cos^2x)^3 = 4 [(1/2) (1 + cos2x)]^3

Expanding this expression, we can simplify it as follows:

4 [(1/2) (1 + cos2x)]^3 = 4 [(1/8) (1 + 3cos2x + 3cos^22x + cos^32x)]

Now, we can simplify further by eliminating the cos^22x and cos^32x terms using another trigonometric identity:

cos^2x = (1/2) (1 + cos2x)

cos^3x = (1/2)^3 (1 + cos2x)^3

So, substituting this into the expression:

4 [(1/8) (1 + 3cos2x + 3cos^22x + cos^32x)] = 4 [(1/8) (1 + 3cos2x + 3[(1/2) (1 + cos2x)] + [(1/2)^3 (1 + cos2x)^3])]

Now, we can simplify this expression further, but it will contain cosines raised to higher powers.