Can I use the whole mathematical proof from references like books and reputed research papers for my own paper? Currently I am trying to publish my research paper on cosmology. I have found some useful proved equations in other research papers. As this is my first research paper, I have no experience with plagiarism in a research paper.

Can I add these proven equations from other papers into my paper?
You certainly can, and should, use the equations (and theorems, etc.) that others have proved. Otherwise, you would have extremely hard time getting anything new done, as mathematics and physics research is quite cumulative in nature.

You use other results by citing the source. At least in mathematics, sometimes you just write "by the Lax-Milgram lemma we get...", if you are writing for an audience which knows what the Lax-Milgram lemma is. If you suppose not all of the audience knows this, then you provide a reference to the result: "by the Lax-Milgram lemma [1, theorem 4.7] we get..." (where the citation format may and does vary). If in doubt, err on the side of citing, as it will be helpful for beginners.

However, if you are writing a research paper and the concept of citations is unfamiliar to you, then it is a good idea to get some help from a scientist in the field (or any field). If you are at a research institute, find someone there. If not, it will be more difficult, maybe impossible, to find a contact, but you might still want to try. Publishing scientific articles is not a trivial task and there are some traps (predatory journals).

[1] Reputable source with bibliography info.