There is a din that before R"H a person has to bathe and take a haircut before Y"T because it's a day of celebration, as we are confident that we will emerge triumphant from standing before Hashem and being judged.
Meforshim ask: The gemara writes that we do not recite hallel on R"H because it would be inappropriate when the sifrei chaim and sifrei meisim are open and our fate hangs in the balance to be celebrating. If we are so confident of a positive outcome, why not celebrate?
To tell you the truth, I am not so bothered by this question. It seems to me that there is a difference between showing confidence and being happy. A lawyer can be confident going to trial, but that doesn't mean he is happy to have to face litigation. We are confident that Hashem will judge us fairly and get us to a positive outcome, but that doesn't mean we should be in a celebratory mood. The mood is still solemn, as becoming the seriousness of being judged. Hallel is therefore out of place.
The Chochmas Shlomo on the SA as well as the Netziv at the end of Emor in the Harchev Davar offer a different answer. On R"H, we actually face two different judgments. Each and every one of us is individually judged as to what kind of year we will have, but in addition, Klal Yisrael as a whole isjudged. Will our enemies have the upper hand, or will we as a people thrive and grow this upcoming year?
The Rama (OC 583) quotes the well known minhag of eating apple dipped in honey. Biur haGR"A explains:
תפוח. ע"ש ופריו מתוק לחכי וכמ"ש כריח שדה ומתרגמינן חקל תפוחים והיה בר"ה כידוע:
It was on R"H that Yaakov went in and got the brachos from Yitzchak (other Midrashim place the date as Pesach), usurping Eisav and making sure we would have the bechora. Chazal (Taanis 29) comment on the pasuk וַיֹּאמֶר רְאֵה רֵיחַ בְּנִי כְּרֵיחַ שָׂדֶה אֲשֶׁר בֵּרְכוֹ ה׳ that Yaakov smelled like an apple orchard, אמר רב יהודה משמי' דרב, כריח שדה של תפוחים. (See Torah Temima on that pasuk who notes that although the GR"A refers to the Targum as the source for it being an apple field, it is not found in the Targum, and GR"A probably meant this derasha).
This was not just a personal victory for Yaakov over his brother, but this is maaseh avos siman la'banim, and it resonates on a national scale. Every year the conflict between Yaakov and Eisav plays itself out on R"H. Every year we and the umos ha'olam face off in judgment, each one of us trying to earn the bracha that will put us on on top. Every year we dip the apple in the honey, assured that we will be the ones to emerge with bracha, just as our forefather Yaakov did.
The gemara writes that R' Abahu in Keisari made a takanah to blow all the different kolos: tekiya-teru'ah-tekiya, tekiya-shevarim-tekiya, and tekiya-sh"t-tekiya. The fact that this takanah was made in Keisari is not just a trivial detail, says R' Chaim Feinstein, but perhaps is linked to the essence of what the day is about. The gemara elsewhere (Meg 6a) tells us:
קסרי וירושלים אם יאמר לך אדם חרבו שתיהן אל תאמן ישבו שתיהן אל תאמן חרבה קסרי וישבה ירושלים חרבה ירושלים וישבה קסרי תאמן שנאמר אמלאה החרבה אם מליאה זו חרבה זו אם מליאה זו חרבה זו רב נחמן בר יצחק אמר מהכא ולאום מלאום יאמץ
Keisari is the stronghold of Eisav, while Yerushalayim is our stronghold. When one is ascendant, the other must fall. Rosh haShana is a replay of the age old conflict between Eisav and Yaakov, between Keisari and Yerushalayim, and so it is davka there that R' Abahu instituted his takana of tekiyos, so as to weaken Eisav.
(Parenthetically, the Rishonim quote BH"G that there was no real doubt that mi'doraysa one is yotzei with any of the combinations of kolos. The R' Abahu made his takanah was to create a unified minhag in Klal Yisrael. You don't have a similar takana, for example, that all of klal yisrael should do nanau'im the same way. Perhaps the reason why is because shofar is linked to this idea of judgment on the tzibur, and so factionalism, having different groups with different practices, would be antithetical to the spirit of the day.)
The poskim write based on Midrashim that the reason we take the lulav and esrog after Yom Kippur is because when an army would emerge victorious from battle they would have a parade and wave their standards and their arms in the air. The lulav is our standard, and we hold in high and wave it around to show that we alone have emerged victorious from the conflict with Eisav.
The mitzvah to bathe and get a haircut, assured that we will have a good outcome on the yom ha'din, is viz a viz the judgment that occurs on a national level. Yaakov every year triumphs over Eisav and remains Hashem's chosen one. (I am still a bit perplexed. So why not say hallel in commemoration of the national victory over Eisav?) As for each one of us individually, what will happen remains to be determined.
That being given, I think one way we can assure ourselves of a positive outcome is to throw our lot in with that of the nation, the tzibur, the klal. This, I think, is what the Isha haShunamit (Melachim 2 4:13) meant when she told Elisha that she does not need any reward because וַתֹּאמֶר בְּתוֹךְ עַמִּי אָנֹכִי יֹשָׁבֶת If instead of focusing on our individual needs and our individual lot we focus on our simply being members of the klal, our judgment will tie in with that of the klal.