Learn Hardcore German: Trotz des Regens bleibt das Lächeln der Kinder breit. - Despite the rain the children's smile remains broad.

Trotz is one of the few German prepositions that traditionally requires the genitive case to express concession (“in spite of”). Therefore you say trotz des Regens (genitive singular of der Regen) rather than trotz den Regen (which would be accusative and is non-standard). Colloquially you might hear trotz dem Regen, but in formal written German trotz + genitive is correct.

Why is it des Regens and der Kinder? How do I know these genitive endings?

German genitive endings vary by gender and number:

Masculine and neuter singular nouns normally add -s or -es:
• der Regen → des Regens
Plural nouns keep the noun form but use the article der for genitive plural:
• die Kinder → der Kinder

So in our sentence:
• des Regens = gen. masc. singular
• der Kinder = gen. plural

Why is the verb bleibt placed right after Trotz des Regens and before the subject?

In a German main clause the finite verb must occupy the second position (V2 word order). When you put the prepositional phrase Trotz des Regens in the first position, the verb bleibt must follow immediately, then the subject das Lächeln der Kinder comes next.

Why do we use bleiben here instead of sein? Could we say das Lächeln ist breit?

Both bleiben and sein can link a subject to a predicate adjective, but they convey different nuances:

bleibt (= “remains”) emphasizes continuity or persistence despite a condition (“despite the rain, the smile stays broad”).
ist (= “is”) simply states a fact about the smile’s width without that nuance of persistence.

You could say Das Lächeln der Kinder ist breit. but it loses the idea that the broad smile endures despite the rain.

Why is breit at the end of the sentence? Can’t I say breites Lächeln der Kinder instead?

Here breit is used predicatively (it describes the state of das Lächeln) and thus follows the verb. If you want an attributive adjective directly in front of the noun, you must decline it and change the sentence structure. For example:
• Trotz des Regens sieht man das breite Lächeln der Kinder.
In that case breite is an attributive adjective and takes the ending -e after the definite article das.

Why is Lächeln capitalized?

In German all nouns are capitalized, and Lächeln is a noun derived from the verb lachen. Any noun—whether a simple object, a concept, or a verbal noun like Lächeln—begins with a capital letter.

Could I use obwohl es regnet instead of trotz des Regens? How do they differ?

Yes, you can. Obwohl is a subordinating conjunction meaning “although” or “even though.” It introduces a full subordinate clause and forces verb-final word order in that clause:
• Obwohl es regnet, bleibt das Lächeln der Kinder breit.
By contrast, trotz + genitive remains part of the main clause with V2 order. Both convey the idea of “despite the rain,” but obwohl always brings in a dependent clause, whereas trotz des Regens is a prepositional phrase.