find an equivalent expression for cos^4 x in terms of function values of x, 2x, or 4x, raised only to the first power

You should use the following:

1 + cos 2x = 2 cos^2 x

Hence Cos^2 x = (1 + cos 2x)/2

NB Two very useful formulae for such problems :

1 + Cos 2x = 2 Cos^2 x

1 - Cos 2x = 2 Sin^2 x