an equivalent expression for cos^4 (x) in a function value of x, 2x, or 4x only to the first power.
Does this work?

cos^4 (x)=[(1+cos2x)/(2)][(1+cos2x)/(2)]
=(1+2cos2x+cos2xcos2x)/(4)

I'm really confused by this problem can you please help me
thank you

I think you are 90% of the way there. You corectly used the identity:

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

Your last expression has the term cos(2x)cos(2x), which is cos^2(2X). So you're not quite done - you want to convert cos^2(2x) into something that does not have a power in it. All you need to do is use the same identity as above, such that cos^2(2x) = (cos2x+1)/2. You should be able to take it from there.