Complex function with specific residues

Question

I am supposed to find a holomorphic function $f:\mathbb{C}\setminus \{-1,+1\} \to \mathbb{C}$ such that it has essential singularities in $-1$ and $+1$ and $res_{-1}f=-1$ and $res_{+1}f=+1$. Is this function unique by these properties?
I can't seem to work it out with both conditions. There are different functions I tried to manipulate so that it satifies the conditions: First $\frac{1}{cosz}$ and second something like $\frac{1}{1-e^{z-1}}$. The first function has the right residue but at the wrong places (acc. to wolframalpha) and I am not even sure if they are essential singularites. I tried manipulating the period in the denominator but it always changes the value of the residue as well. On the other hand The second function already has at least the singularity $+1$ and the right value, but if I add another term for $-1$ it breaks. The $cos$ function has the advantage that I would already know, that there is some sort of sin-function with the same properties, so it wouldn't be unique. In general I think there could be moutiple functions with these properties.
Any idea how to work on this more systematically?