The `GenerateConditions->True` option of the `Integrate` command is useful
 and works properly in most cases. However, there are cases when this command
works improperly. Here is one of these: the result of

    Integrate[1/(1 - Cos[x]), {x, a, b}, Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]

> `ConditionalExpression[Cot[a/2] - Cot[b/2], a^2 < a b || a b > b^2]`

is not correct if the integration range includes a zero of `Cos[x]-1`.

The result of

    Integrate[1/(1 - Cos[x]), {x, Pi/4, b}, Assumptions -> {b} \[Element] Reals, GenerateConditions -> True]

> `-Cot[b/2] + Cot[\[Pi]/8]`

is not better. The same issue in 12.3.1 on Windows 10 with `Integrate[1/(Sqrt[3] - 2*Cos[x]), {x, a, b}, 
 Assumptions -> {a, b} \[Element] Reals, GenerateConditions -> True]` etc.

Is there a workaround?