It is possible to write $$ \sqrt{ 2 + \cos(x) + \cos(y) } -2 = \frac{-a(x,y)\sin(x)}{\sqrt{ 2 + \cos(x) + \cos(y) }} + \frac {b(x,y)sin(y)}{\sqrt{ 2 + \cos(x) + \cos(y) }}$ $$ for some functions $a,b \in C^1(-\epsilon,\epsilon)$?
To give some context. Let $f = \sqrt{ 2 + \cos(x) + \cos(y) } -2$ and $\omega = dx \wedge dy$. Is there some 1-form $\alpha$ such that $ df \wedge \alpha = f \omega $ locally?
I infer from your conditions on $a,b$ that by "locally" you mean only in a neighborhood of $(0,0)$. This point is a local maximum of $f$ so, in fact, both $f$ and the $1$-form $df$ vanish at $(0,0)$. And $f$ is real analytic in a neighborhood of $(0,0)$, so you can write a power series expansion for $f(x,y)$ in a neighborhood of $(0,0)$. Indeed, using well-known expansions for $\cos(u)$ and $\sqrt{1+u}$, we see that $f(x,y) = \phi(x)+\psi(y)$, with $\phi(0)=\phi'(0)=\psi(0)=\psi'(0)=0$. Writing $\alpha = b(x,y)dx+a(x,y)dy$, as you did above, just set $b(x,y) = -\psi(y)/\psi'(y)$ [and this has a well-defined power series expansion at $0$] and $a(x,y) = \phi(x)/\phi'(x)$.