Is there any hidden rule for using the words "clearly", "obviously" or similar ones in a technical paper? It can be offensive to the readers in many cases (especially in mathematical proofs), since the reader may not find it "clear" or "obvious". But does that mean that we should completely avoid the use of these words?
Seconding posdef's appraisal, but being a little more blunt: if one is in a position to get away with bullying or intimidating people by implying that it's their problem if one has not explained well enough ... well, I'd say it's still a jerk-y thing to do. If one is in a lower-status position, such words will often be red flags.
Or, coming to functionality versus rhetoric versus "formal proof": at best these words are functionless filler. That is, saying something is clear is not what makes it clear: if it is clear after these words, it was clear before. Conceivably a thing is clear _once_noted_, and thus deserves "Observe that...". But this, too, can be abused if used outside situations where one is noting that something is "a-fortiori" true, that is, is weaker than what the argument has already demonstrated... but presumably suffices for the issues at hand.
I don't think there is a very clear rule for using such words. One possible reason for my claim is that some authors don't even use words "clearly" or "obviously", but they simply say "it follows ...". In mathematics the level of details of a mathematical proof mostly depends on the writer's kindness to her/his readers. I have encountered with many not-so-obvious claims in papers written by experts, where needed several pages of explanations and perhaps some proofs, and several years later, I have found the proofs of those claims in newer papers written by other authors.
Unfortunately, there is an adage which says "brevity is a sign of genius" and it seems some people strongly believe in this adage and try to impress others by leaving not-so-obvious gaps in their works.
Personally I apply the following rules for using these words:
If the claim follows from previously mentioned materials by applying well known techniques in 5 minutes or so.
If it can be obtained by a few lines of computations again by applying well known techniques. Then I use the word "straightforward".
If it easily follows from a well known type of mathematical proofs, like induction, Zorn's lemma.
The proof is similar to a previous proof in the paper or in the literature. In this case I mention the resource.
I expect a PhD student in the field can prove it easily.
I don't think there is a clear consensus on how to use these words.
As mentioned in some other answers, some people find them annoying or obnoxious. Others think they are a perfectly acceptable way to mention a fact for which you believe a detailed explanation is not necessary. Certainly they are quite common in published writing.
I think it is a choice that you make as part of developing your own personal writing style, and your feelings may change over time.
My only advice is: when you write that something is "obvious", make absolutely sure it is true! I've been embarrassed this way before.
More broadly then in regards to mathematical proofs, a mark of good writing is to avoid the superfluous. Whether something is clear or obvious comes from the content, not the writer labelling it as such. Trimming unneeded adjectives and adverbs like those you describe should be a regular step in a proof-reading stage. See Strunk and White's Elements of Style for a more detailed treatment.
We touched this particular subject in a "Technical Writing" course; the simple answer is that it's a power-stance. In other words, if you are a big-name professor in your field, you can use it without offending someone. Alternatively if you are a petty PhD candidate, then you are better off avoiding not only these two words but also other forms of bold statements when you are drawing conclusions.
As I said this is rather the short answer, I am sure those who are more into linguistics etc might have more insight into the matter.
By reading the comments and answers here, the conclusion is, that it is usually not a good idea to use these terms. Keep in mind that it might not always be the case that something is obvious to your reader. That being said, the reason you want to use such words is probably because you want to point out/conclude/summarize your findings to the reader.
The bottom line is not to tell your readers what (you find) is obvious, but to tell them what the obvious thing is (conclude/summarize). This way they will either:
A. Confirm their own observation
B. Let them know they haven't fully understood yet (they might re-read your article now)
I propose never using these words unless your goal is to trick the reader into thoroughly checking your claim, or in an exam's trick question where you set a false premise (though these words are give-aways if not overused). If something is obvious there wouldn't be a need to even state it. And if you need to state something, it is not obvious.
If you think some non-trivial1 steps should be omitted so your 5 page paper doesn't bloat up to a 30 pager, then please have the decency to either briefly state the trickiest tool involved (be that induction or some specific part of Wile's proof of Fermat's Last Theorem) or - even better - put the detail which you should have done anyway into the appendix / online supplement and refer to it.
1Trivial is also one of these words.
I might go against most of the answers here and say why not?.
I am going to this right now. I am writing a paper proposing a solution for problem X by adopting well known mathematical model Y. Now Y has clear axioms and definitions (for instance, the set of considered elements has to form a commutative semigroup under combination). I defined X then defined the combination operator. Should I go further and proof it is commutative semigroup? I believe it is clear that X form a commutative semigroup within my framework. Yes It is obvious..
Now whether the author of these words is a student or professor, I believe it doesn't make difference. At the end, there is minimum knowledge required to understand any given paper, if its clear then it's clear and you better utilize the paper limited space in something not clear enough.