TITLE: Why are math texts usually so compact?
QUESTION [11 upvotes]: I'm sorry if this isn't an appropriate place to ask this.
I taught a couple years in a secondary school, where teachers are supposed to explain every little step in the simplest but accurate way.
In high school things don't change much. Perhaps the pace is a little faster but still the goal is to be as clear as possible. Teachers are supposed to use pictures, animations and justify every step when solving problems.
In the university things are a lot different. We get exercises every week and solutions are extremely compact. I don't mean to show every little step when computing a determinant of a $3\times 3$ matrix but often one reads "one implication is trivial, one can (easily) see that, it is clear that..." and so on, which makes me personally feel incompetent because most of the times I don't easily see what I'm supposed to. At least in secondary school, it is mandatory to avoid using words such as "trivial, easy" and so on because of that reason. Still, some teachers do it and some believe this to be one of the causes of the spread of anxiety for mathematics, which certainly hinders the learning process even more.
This applies to this platform as well. The style completely changes.
Why is that so? Are compact solutions more elegant? Are mathematicians just lazy?
I'm very curious about it.

REPLY [8 votes]: This is more of a comment than an answer and might be seen as some sort of addendum to J.G.‘s answer.
It is completely correct that in school one is trained to follow algorithms, while in university the focus switches to writing correct algorithms (=proofs). Moreover it is very tempting (if not necessary in the case of research papers for example) to gloss over standard details to get to more interesting stuff.
Yet I feel like using words like trivial, obvious or easy is very bad practice, especially in the context of mathematical education. It unnecessarily makes students feel bad, if they don’t get the detail, sometimes even after hours of staring at it. It always is a triviality at the end, but most of the time it needs that one standard trick, which just isn’t memorized yet.
I don’t understand, why it isn’t more common to give concise sketches of the idea required to prove a result:

by definition says plugging in the definitions will make the result obvious
by iteratively using lemma X on Y is a pretty concrete recipe
by strong induction on $n$ gives information about which type of induction hypothesis is needed

et cetera, et cetera. I don’t think this requires that more time and space and most importantly it gives some sort of reality check: If you cannot summarize the triviality in like one sentence, it most likely is not so trivial at all...