legitimieren
to legitimize
To recognize or declare something as lawful or justified; to create a legal or moral basis for something.
Die Wahl hat die neue Regierung legitimiert.
The election legitimized the new government.

to legitimize / to identify oneself
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to legitimize
To recognize or declare something as lawful or justified; to create a legal or moral basis for something.
Die Wahl hat die neue Regierung legitimiert.
The election legitimized the new government.

'Legitimieren' (to legitimize) is derived from the adjective 'legitim' (legitimate, lawful, justified). So the verb means to recognize something as legitimate or to prove one's own legitimacy.
Durch seine guten Argumente konnte er seine Position legitimieren.
With his good arguments, he was able to legitimize his position.
When it is about proving one's own identity, the verb is always used reflexively: 'sich legitimieren' (to identify oneself). The subject and the reflexive pronoun are the same person.
Können Sie sich bitte mit einem gültigen Dokument legitimieren?
Could you please identify yourself with a valid document?
When proving identity, the preposition 'mit' (with) and the dative are often used to name the document. Typical patterns are 'sich mit einem Ausweis/Pass legitimieren' (to identify oneself with an ID card/passport).
Am Eingang musste ich mich mit meinem Presseausweis legitimieren.
At the entrance, I had to identify myself with my press ID.
In the meaning 'rechtfertigen' (to justify), 'legitimieren' (to legitimize) is often used with abstract objects in the accusative. Common combinations are to legitimize 'eine Handlung' (an action), 'eine Entscheidung' (a decision), or 'eine Machtposition' (a position of power).
Der König versuchte, seinen Anspruch auf den Thron zu legitimieren.
The king tried to legitimize his claim to the throne.
to justify / establish
'begründen' and 'legitimieren' can both explain why something appears justified. 'begründen' is used when you give reasons and make a decision, opinion, or action understandable. 'legitimieren' is stronger and often more formal: it is about presenting something as lawful, permitted, or morally justified. You can justify a decision with arguments; you legitimize a measure when it is meant to be seen as justified because of them.

Die Regierung versucht, die neuen Kontrollen mit dem Schutz der Bevölkerung zu legitimieren.
The government is trying to legitimize the new checks with the protection of the population.
Die Chefin begründet die Änderung mit den vielen Beschwerden der Kunden.
The manager justifies the change with the many customer complaints.