sich weiten
to widen
To become wider, larger, or more spacious. This often refers to openings, vessels, items of clothing, or pupils.
Seine Pupillen weiteten sich vor Schreck.
His pupils widened in fright.

to widen / broaden
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to widen
To become wider, larger, or more spacious. This often refers to openings, vessels, items of clothing, or pupils.
Seine Pupillen weiteten sich vor Schreck.
His pupils widened in fright.

The verb 'weiten' (to widen) is derived from the adjective 'weit' (wide, spacious). It describes the process of becoming wide or actively making something wide.
Durch das viele Tragen haben sich die Schuhe etwas geweitet.
The shoes have widened a little from being worn so much.
This verb is often used reflexively ('sich weiten' — to widen) when something becomes wider on its own. But it can also be used transitively with an accusative object when someone actively makes something wider.
Bei Dunkelheit weiten sich die Pupillen automatisch.
In darkness, the pupils widen automatically.
The reflexive form 'sich weiten' (to widen) is often used in biological or medical contexts, for example with pupils or blood vessels. It also describes how a landscape or a space opens up.
Als sie aus dem Tunnel fuhr, weitete sich das Tal vor ihr.
As she drove out of the tunnel, the valley widened before her.
In the transitive form, 'weiten' (to widen) is often used in connection with clothing or shoes that are too tight. Typical objects are trousers, rings, or shoes.
Ich muss dieses Hemd am Kragen etwas weiten lassen.
I need to have this shirt widened a little at the collar.
The phrase 'den Blick weiten' (to broaden one's view) is a fixed expression. It is used metaphorically and means to broaden one's horizons or become more open to new things.
Reisen hilft dabei, den eigenen Blick zu weiten.
Traveling helps broaden your own view.
to expand
'erweitern' and 'weiten' can both mean that something becomes larger or is made larger. 'erweitern' fits very well when an area, an offering, a building, or a plan becomes larger, often through something new. 'weiten' is used more when something is made wider or looser, for example clothing, an opening, or even one's view. Learners may confuse the words because both have to do with 'larger'. As a guide: you use 'erweitern' for an offering or knowledge; you use 'weiten' for a tight shoe or one's view.

Der Schuster weitet die Schuhe, weil sie vorne zu eng sind.
The shoemaker widens the shoes because they are too tight in the front.
Die Firma erweitert ihr Angebot ab Herbst um neue Kurse.
The company is expanding its offerings with new courses starting in the fall.