trocknen
to dry
To make something wet or damp dry by removing the moisture from it.
Nach dem Duschen trockne ich mir die Haare mit dem Föhn.
After showering, I dry my hair with a hair dryer.

to dry
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to dry
To make something wet or damp dry by removing the moisture from it.
Nach dem Duschen trockne ich mir die Haare mit dem Föhn.
After showering, I dry my hair with a hair dryer.

The verb 'trocknen' (to dry) is derived directly from the adjective 'trocken' (dry). When something 'trocknet' (dries), it becomes dry.
Die Farbe an der Wand ist endlich trocken.
The paint on the wall is finally dry.
This verb can be used in two ways. Either a person actively dries something (transitive with an accusative object), or a thing dries on its own (intransitive without an object).
Er trocknet das Geschirr ab.
He dries the dishes.
When something dries on its own, the perfect tense can be formed with both 'haben' and 'sein'. 'Sein' places more emphasis on the completed process and the state that has been reached.
Die Wäsche hat in der Sonne schnell getrocknet.
The laundry dried quickly in the sun.
The idiom 'jemandem die Tränen trocknen' (to dry someone's tears) has a figurative meaning and means 'jemanden trösten' (to comfort someone). The person being comforted is in the dative.
Nach der Trennung haben ihre Freunde ihr die Tränen getrocknet.
After the breakup, her friends dried her tears.
to calm
'beruhigen' and 'trocknen' can overlap in one particular situation: when someone is crying. 'beruhigen' directly says that you help an upset or sad person become calmer again. 'Tränen trocknen' (to dry tears) is more figurative and means to comfort someone who is crying. It puts more emphasis on the crying and the comfort. In everyday use, 'beruhigen' is more general and also fits with fear or anger. 'Tränen trocknen' only fits when sadness or crying is really the focus.

Nach der traurigen Nachricht trocknet ihre Freundin ihr die Tränen.
After the sad news, her friend dries her tears.
Der Vater beruhigt das Kind, nachdem es sich erschrocken hat.
The father calms the child after it has been frightened.