Solved
As shown in the other thread both are roots and there are many others.
Where do the 1.835... stem from?
Adding pi to get another root is not correct because your function is not periodic. It may look that way for small (negative) values of x because e^(x-pi) gets rather small then. Nevertheless the roots are just close to the roots of cot(x) but always are a bit to the right of them.

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