Generally speaking, women are exempt from the Musaf prayer recited on Shabbat, Rosh Hodesh and festivals, and they should not recite this prayer even if they wish to. On Rosh Hashanah, however, the Musaf prayer includes numerous verses and passages relating to the theme of divine mercy and our prayers for a good year, and thus women may recite Musaf on Rosh Hashanah. In fact, according to some views, women are obligated to recite Musaf on Rosh Hashanah, as well as on Yom Kippur. It goes without saying that women may recite the Ne’ila prayer on Yom Kippur.
Even though Hallel is normally recited on Yom Tob, we do not recite Hallel on Rosh Hashanah or Yom Kippur. As these occasions are days of judgment when the books of life and death are opened before God, it is inappropriate to conduct the joyous reading of Hallel. And although we feel confident on Rosh Hashanah that we will be inscribed for a good year, we must nevertheless approach our judgment with fear and dread, emotions which are incompatible with the singing of Hallel.
However, if a person recites Tehillim on Rosh Hashanah, and over the course of his reading he reaches the chapters of Hallel, he may recite these chapters. Since he recites these chapters as part of his normal reading of Tehillim, and not in the context of the festive Hallel recitation, this is permissible.
Summary: Women may recite Musaf on Rosh Hashanah, and, according to some opinions, they are required to recite Musaf on Rosh Hashanah. Hallel is not recited on Rosh Hashanah or Yom Kippur, though one who recites Tehillim may recite the chapters of Hallel as part of his Tehillim reading.
