I've been using a learning app and this sentence came up, Willst du ins Kino gehen? and the translation is Would you like to go to the cinema?.
For all other times möchte has been used for would like. Is willst correct in this example?
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In most cases, the forms möchtest du, magst du, willst du can be used interchangeably without changing the meaning.
The difference is style. Many parents attempt to teach their children to use möchte rather than will because it is more polite especially if the subject is I:
Ich will ins Kino gehen.
Ich möchte ins Kino gehen.
You can compare this to ‘I want to go to the cinema’ and ‘I would like to go to the cinema.’ Mögen in its indicative mood is somewhere in-between but has fallen out of use for this time of expression; rather, it is now used to express enjoyment.
For completeness: möchte is not a verb by itself but the conjunctive II form of mögen.
The literal translation of
Would you like to do something?
Would you like to sleep?
Would you like to go to the cinema?
Würdest du es mögen, etwas zu tun?
Würdest du es mögen, zu schlafen?
Würdest du es mögen, ins Kino zu gehen?
But the phrase »würdest du es mögen« is no good style. It is grammatically correct, but it is bumpy and awkward German.
A better phrase with the same meaning, but of a much smoother style is »möchtest du«:
Möchtest du etwas tun?
Möchtest du schlafen?
Möchtest du ins Kino gehen?
(As you can see, now there is no »zu« before the infinitive!)
The verb »möchten« has a synonym, which is »wollen«. Both tranlate into the english verb »to want«.
So you also can say:
Willst du etwas tun?
Willst du schlafen?
Willst du ins Kino gehen?
Be aware, that »Ich will« does not translate into »I will«! It is »I want to«! (see also https://german.stackexchange.com/a/25474/1487)