I can't really think of a philosophical example, but in mathematics, it would be something like 'find the x where f(x) = 0 given that f(x) = x', and the solution would be 'x = 0', which is trivial. Is there any such "theorem" in philosophy? What are the most popular ones? Is there even an analog?
The answer is yes and no. Philosophical theory itself does not because philosophical texts are largely exercises in defeasible reason conducted in natural language which is far more syntactically and semantically complex than artificial language. However, formal logic, like problems conducted in Fitch, can have trivial solutions.
The question isn't quite what would be considered well-formed which means there are some usages in your language that reflect the need to explain relationships between philosophy, mathematics, and logic.
Philosophy is chiefly conducted through argumentation, and those arguments are primarily divided between formal logic and informal logic with the latter being more encompassing and more related to natural language. As such, philosophical discourse doesn't have "trivial solutions" in the same way mathematical theory might.
Philosophy, generally speaking, doesn't use theorems which by this article is a mathematical term:
In mathematics, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems.
That being said, philosophy deals almost exclusively in propositions which is a unit of meaning roughly based on a sentence, but somewhat flexible in syntax. For instance, two different sentences can represent the same proposition:
The boy is holding the blue ball.
The boy is holding the ball that is blue.
Both of these would be considered relatively identical in meaning. A mathematical theorem doesn't manifest such syntactic flexibility and therefore is not open to debate in meaning. Natural language propositions are and that's why philosophy (and law and argumentation) are seen as exercises in hair-splitting. Formal logic shares the same nature and as such, within formal logic, there are trivial solutions to formalisms. A simple example would be determining the truth condition of the statement p=p which is true by the law of identity.
So, it's fair to say that the trivial solutions of formal logic have a similarity to those of mathematics in that the problems are generally simple and reduce to the axioms of the formal system with relative ease requiring few if any inferences.