Help with addition formula question

...

You initially had a "cos x" on the LHS (correctly), but crossed it out for some reason and treated it as zero. Three lines up.

Adding back the cosx only leads me up to
$2cosx = \sqrt{2}cosx - \sqrt{2}sinx$
and then I am stuck again

Adding back the cosx only leads me up to
$2cosx = \sqrt{2}cosx - \sqrt{2}sinx$
and then I am stuck again

Adding back the cosx only leads me up to
$2cosx = \sqrt{2}cosx - \sqrt{2}sinx$
and then I am stuck again

It's from Pearson's Advanced Maths A2 Core for Edexcel
pg65 Ex3C Q8b

Solve for values of x between 0° and 360°

cos(x + 45°) = cosx

I've tried to work it out but I get stuck and we didn't do a question similar to this in class to compare it to

This is my working so far but it doesn't leave me much to go off so I presume it's incorrect.

I think the simplest way to use that rule when

$\displaystyle \cos A =\cos B$

then the 2 cases are:

1: $\displaystyle A=B + 2k\pi$
or
2: $\displaystyle A=-B + 2k\pi$

or A=B?