I don't think they're synonyms, but responsible can convey multiple meanings, like how in German there's both verantwortlich and zuständig - the former meaning something like you are responsible for your actions and the latter meaning something like you are a reliable person.
Slight correction to your German: verantwortlich and zuständig both mean responsible and there is very little difference between the two words. Verantwortlich means you carry the responsibility. If something goes wrong you get the blame. Zuständig is a legal term. You are the one to do the job and no one else. The word for saying that you are a reliable person is "verantwortungsvoll" (lit. full of responsibility)
Trustworthy and responsible, in English, are considered synonyms of one another and not all that loosely related. At www.thesaurus.com, it shows you the relative strength of the closeness in meaning of synonyms through shading. The darker the shading is, the stronger the connection.
That said, the word “responsible” is used far more frequently than “trustworthy.” Not too surprising since "responsible" is a word that encompasses more nuances of meaning than "trustworthy." Be that as it may, the use of the word “trustworthy” is on the rise while the use of the word “responsible” has dropped a bit in recent years. I know that just about any resumé writing class I've ever attended advises against the use of "responsible," encouraging job seekers to use "strong action verbs" instead. Whether or not that has anything to do with the drop in the use of "responsible" is anyone's guess, but a lot of people do post their resumés online now! Then again, perhaps the world we've lived in during the past few years has not given English speakers as many reasons to use the word as it has in years of the more distant past. It would be interesting to see if its equivalent in other languages is rising or declining and whether or not such a trend reveals something about society and what its impact might be. Regardless, the word "responsible” has been in use for far longer than “trustworthy" and that, too, must have something to do with it being a more frequently used word, despite any decline in its usage.
Being two different words of course, they do have different meanings. First meaning listed for trustworthy is “deserving of trust or confidence.”
First meaning listed for “responsible” is “answerable” or “accountable.”
One last thing. Though “trustworthy” and “responsible” are fairly strong synonyms in English and I would suspect most English speakers use them interchangeably without much thought as to whether “deserving of trust” or “accountable” is meant, they are both different enough to be used in a single sentence without being considered too repetitive. If those are the only two words you’re using to describe a person or a character, however, your description isn’t likely to be considered all that detailed, comprehensive, or enlightening but that type of discussion might be a bit too OT for duolingo at this level. Nevertheless, the decisions a person makes in word choice should have a solid basis in semantics, and the thinking that goes into making those decisions does contribute to becoming more fluent in a language. So, I tip my hat to duolingo for giving us this opportunity to discuss nuances in words more fully and appreciate those who regularly take the time to do so.
In my intuition, "responsible" (as a character trait) and "trustworthy" have slightly different shades of meaning, where "trustworthy" focuses on the person's capability for a particular task, whereas "responsible" is more of a general "getting things done that need to be". Note, of course, that each has meanings not synonymous with the other. I can use "trustworthy" as a synonym for "honest", but not so with "responsible". Also, "responsible" has another meaning that selects an argument ("he is responsible for this task" vs "*he is trustworthy for this task").