
zugeben
to admit
To acknowledge or confess a mistake, guilt, or fact, often reluctantly.
Schließlich musste er zugeben, dass er sich geirrt hatte.
In the end, he had to admit that he had been mistaken.
to admit / to add
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to admit
To acknowledge or confess a mistake, guilt, or fact, often reluctantly.
Schließlich musste er zugeben, dass er sich geirrt hatte.
In the end, he had to admit that he had been mistaken.

to add
To add something to a mixture or a dish.
Für den Teig müssen Sie noch zwei Eier und etwas Milch zugeben.
For the dough, you still need to add two eggs and some milk.
'Zugeben' (to admit/to add) is a separable verb. In a main clause, the prefix 'zu' goes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Er gibt seinen Fehler offen zu.
He openly admits his mistake.
When you admit what you did or what is the case, a subordinate clause with 'dass' often follows. The conjugated verb of the subordinate clause then comes at the end.
Sie gab zu, dass sie das Dokument verloren hatte.
She admitted that she had lost the document.
In the meaning of 'hinzufügen' (to add), 'zugeben' (to add) is used almost exclusively in instructions such as recipes or chemical processes. So the context is cooking, baking, or mixing.
Geben Sie zum Schluss noch eine Prise Salz zu.
At the end, add a pinch of salt.
to recognize / acknowledge
Both words can express that someone accepts an unpleasant fact. Anerkennen emphasizes that one now accepts the truth of it. Zugeben more strongly emphasizes openly admitting something that one had previously concealed or denied.

Nach langem Leugnen gab Paul zu, dass er die Zahlen absichtlich verändert hatte.
After denying it for a long time, Paul admitted that he had deliberately altered the figures.
Nach der Prüfung der Zahlen erkannte Paul an, dass seine Berechnung falsch war.
After reviewing the figures, Paul acknowledged that his calculation was wrong.