1a) I am bored to go to the office.
this sentence doesn’t make sense, it’s not proper syntax
1b) I am bored having to go to the office.
this sentence is more correct, and placing “having to” does change the meaning, but it is a weak phrase, and it’s not really “having to” that’s the problem, but rather “to go”. It’s a pseudo ‘be verb’ and there’s a better way to say it, eg: “Going to the office is boring”.
2a) How to look pretty without having to wear makeup?
This sentence is incorrect. Firstly, it ends with a question mark but it is not a question. At least not properly constructed. Again it contains be-verb “to wear”, but that doesn’t help matters, nor change the meaning of (2b). This sentence sounds like the title of an article in Cosmo magazine, not an actual question.
2b) How to look pretty without wearing makeup?
This isn’t any better than 2a. The best way to say this is to replace “to” with a pronoun, such as “How I look pretty without makeup”, but even then it is a fragment, requiring a colon followed by the explanation of how you look pretty without makeup. If you really wanna put that question mark to use properly, it would be “How can I look pretty without wearing makeup?”.
3a) You can sell arms without always having to use them.
Technically a correct statement, but again the weak “to use” is followed by the clunky “having” verb. For best results, and the application of strong verbal voice, we would probably be better off going with (3b), which hasn’t changed the meaning much at all.
Vs
3b) You can sell arms without always using them.