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.