trigonometric again help help!!!

I think you want to prove
cos^4a = 3/8 + 1/2 cos(2a) + 1/8 cos(4a)
Since
cos(2a)=2cos^2a-1
cos(4a)=2cos^2(2a)-1=2[(2cos^2a-1)^2]-1=2(4cos^4a-4cos^2a+1)-1
=8cos^4a-8cos^2a+1
RHS
=3/8 + 1/2 cos(2a) + 1/8 cos(4a)
=3/8 + 1/2 (2cos^2a-1) + 1/8 (8cos^4a-8cos^2a+1)
=3/8 + cos^2a-1/2 + cos^4a-cos^2a+1/8
=cos^4a
=LHS