Given that sin(x) + cos(x) = 2/3, find cos(4x)

It is not clear what to do when starting, but we realise that by making both sides to the power of two, will lead to have an expression containing sin^2(x) + cos^2(x), which is equal to one (which will probably make things easier):
sin^2(x) + cos^2(x) + 2sin(x)cos(x) = 4/91 + 2sin(x)cos(x) = 4/9 2sin(x)cos(x) = -5/9
We also know thanks to double angle identities that 2sin(x)cos(x) is just sin(2x) so we just substitute:
sin(2x) = -5/9
Now we have an expression in terms of sine, but we want it in terms of cos. We take a look at the formulas for double angles (double angle identities) and find out that cos(2k)= 1-2sin^2(k). It is fairly straight forward to solve from this point but to make it simpler, it is useful to make k=2x. So thatcos(4x) = 1-2sin^2(2x)
We know what sin(2x) is socos(4x) = 1- 2* (25/81)cos(4x) = 31/81