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Shemini Aseret is a holiday following Sukkot and has a completely independent nature from the holiday of Sukkot. 
Prayer on Shemini Aseret
- It is customary to recite Tehillim 12 which begins "Lamnatzeach al hashiminit" on the night of Shemini Aseret. 
- Because Shemini Aseret is a separate holiday, if during the Shemini Aseret prayers, a person mistakenly said in the "ata bichartanu" section "et yom chag hasukkot hazeh" instead of "et yom shemini chag aseret zeh", and he continued praying the proper yom tov prayer, and did not realize his error until the middle of the beracha of "ritzeh", or even after "sim shalom", or even in the middle of "elokay netzor", one must return to "ata bichartanu" and correct one's mistake. 
- If one finished the amidah and stepped back, one must return to the beginning of the amidah and start over. 
- If one is unsure as to whether he said the correct version of the amida, one should ideally stipulate the following: "if I am obligated to repeat the amidah over again, then I am doing so as I am supposed to. And if I am not obligated to, then this repetition of the amidah should be considered a "voluntary prayer".
- It makes no difference for our purposes if one actually knew it was Shemini Aseret and just mistakenly said the wrong words, or if the cause of his error was that he did not know that it was Shemini Aseret. In all of these cases if he did not mention Shemini Aseret correctly, he must repeat the amidah. 
- On Shemini Aseret outside Israel the Torah reading is Kol Bechor in Parshat Reah. If Shemini Aseret falls out on Shabbat, the torah reading begins from Asar Taaser. 
Lulav and Etrog
- We do not take the 4 minim on Shemini Atzeret.
Eating and Sleeping in the Sukkah
- Outside Israel, on the eighth day of Sukkot (first day of Shemini Aseret), one should eat in the Sukkah without a Bracha of Leshev BaSukkah. There’s a minhag to be lenient regarding eating in the Sukkah on Shemini Aseret, however, one should not follow this unless one has accepted this practice from his parents or rabbis. 
- Regarding sleeping in the Sukkah on the eighth day of Sukkot (Shemini Aseret), there’s a dispute in the achronim. The Ashkenazim minhag is not to sleep in the Sukkah, while Sephardim hold that one must sleep in the Sukkah. 
An American in Israel
- A Jew from outside Israel who is visiting Israel for Sukkot should not eat in the Sukkah if it’s difficult to sit alone in the Sukkah.  However, if he is alone anyway, such as if he’s in a hotel, Sephardim holds that he should eat in the Sukkah, while Ashkenazim hold that in Israel one doesn't sit in the Sukkah on Shemini Aseret. 
- In Israel one does not eat or sleep in the Sukkah on Shemini Aseret (as it is no longer Sukkot). 
- In Israel, if one must eat or sleep in a Sukkah on Shemini Aseret as there’s no room inside, then one should remove 4x4 Tefachim of S'chach to show that one doesn’t intend to add to the mitzvah (Baal Tosif). 
- An alternative to removing S'chach is to put a tarp on top or below 4x4 Tefachim of S'chach as long as the tarp is within 3 Tefachim of the S'chach. 
- It’s permissible to eat in the Sukkah after Shemini Aseret and it’s not an issue of Bal Tosif. 
- It’s permissible to eat or sleep in the Sukkah before Sukkot (erev Sukkot) and it’s not an issue of Bal Tosif. 
- Because Shemini Aseret is an independent holiday, the beracha of Shehecheyanu must be recited at kiddush. 
- If someone mistakenly said the kiddush for Sukkot instead of the kiddush for Shemini Aseret, he must return and repeat the kiddush. 
- Many poskim say that one shouldn't start the meal until it is certainly nightfall (i.e. Tzet HaKochavim). However, after the fact, if one did start the meal before one should not make the bracha of Leshev BaSukkah if one is eating in the Sukkah. 
- Similarly, one should not make Kiddush on the night of Simchat Torah before certain nightfall (i.e. Tzet HaCochavim). 
- Sitting in Sukkah on Shemini Atzeres by Rabbi Aryeh Lebowitz
- Gemara Rosh Hashana 4b and Sukkah 48a.
Rashi (Vayikra 23:36) states that the reason for the eighth day, Shemini Atzeres, can be explained with the parable of a king who invited his children for several days of feasting. When the time came for them to leave, the king said, "Your departure is so difficult for me. Please stay with me for yet one more day." After seven days of Succot God asks us to stay with Him for yet one more day
- Chazon Ovadia Sukkot p 451
- Chazon Ovadia Sukkot 451-454
- Chazon Ovadia Sukkot 451-454
- Chazon Ovadia Sukkot 451-454. See there where Hacham Ovadia quotes many who agree with him among them Maran Hachida, Rav Chaim Palaggi, Ma'amar Mordechai, Maharsham in the name of Rav Akiva Eiger, S'deh Chemed, Shoel Umeshiv, and Rav Moshe Feinstein, and some who disagree, among them the Kaf Hachaim Sofer, Chayeh Adam, Bet Yehudah Ayash, Rav Shlomo Zalman Auerbach, and the Tzitz Eliezer, but it seems from there that majority of the Acharonim say like this, for both Sephardim and Ashkenazim.
- The gemara Megillah 31a writes that the torah reading for Shemini Aseret is Kol Bechor. Shulchan Aruch 668:1 codifies this halacha. Kaf HaChaim 668:17 explains that we read Kol Bechor because it includes a hidden reference to Shemini Aseret in the words VeHayita Ach Samech which chazal explain to mean that Shemini Aseret is included in the Simcha of Sukkot.
- Gemara Sukkah 47a. Tosafot d"h Maitav explain that the eighth day of Sukkot differs from the eighth day of Pesach in that the eighth day of Sukkot is Shemini Atzeret, which is a Yom Tov that in many ways enjoys a separate and distinct identity from Sukkot. Were we to observe the Mitzvot of the seventh day of Sukkot on Shemini Atzeret, we would undermine the integrity of Shemini Atzeret. We do not take the Lulav on Shemini Atzeret, because the Lulav is ordinarily Muktza. Thus, taking the Lulav on Shemini Atzeret would make a blatant statement of concern that it is actually the seventh day of Sukkot. The Kesef Mishneh (commenting on Rambam Hilchot Sukkah 6:13) offers a different explanation for why we do not take a Lulav on Shemini Atzeret. He notes that it is only a rabbinical obligation to take the Lulav on the seventh day of Sukkot. Biblically, we are only obligated to take the Lulav on the first day of Sukkot if one is outside of the Bait Hamikdash and seven days if one is inside the Bait Hamikdash (see Vayikra 23:40 and Sukkah 41a). The Kesef Mishna argues that the obligations of Yom Tov Sheni do not apply to Lulav because the obligation is only rabbinic in nature
- Rambam (Sukkah 6:13), Tur and S”A 668:1 write that one should eat in the Sukkah on the eighth day of Sukkot but not make a Bracha. Kitzur Shulchan Aruch 138:4, Shu"t Yechave Daat 2:76 agree. This is based on the Gemara (Sukkah 47a) which presents a debate among the Amoraim regarding how residents of the Galut should conduct themselves in relation to Sukkah on Shemini Atzeret. The Gemara concludes that the proper practice is to sit in the Sukkah on Shemini Atzeret but omit the Beracha. The Gemara indicates that we do not take the Lulav on Shemini Atzeret. Tosafot 47a d"h maitav explain that reciting the Beracha would undermine the status of Shemini Atzeret by blatantly demonstrating that we are concerned perhaps the day is not truly Shemini Atzeret.
- Even though the pasuk in Parashat Emor (Leviticus 23:42) states בסוכות תשבו שבעת ימים, meaning you shall dwell in sukkot for seven days, we sit in the sukkah out of the doubt that it may be the 7th day of sukkot which would require sitting in a sukkah.
- There’s an old minhag to be lenient not to eat in the Sukkah on Shemini Aseret from the time of Rashi recorded in Sefer HaPardes, Machzor Vitri (Siman 284), and Maharil (Hilchot lulav) (see nitei Gavriel (Arba Minim) Sh”t 4). The Piskei Teshuvot 668:1 brings in the name of the achronim that there’s room to be lenient since the Korban Netanel writes that the entire establishment of sitting in the Sukkah on the eighth day is based on the assumption that it’s comfortable and beloved to a person, however, if it’s cold or wind is present then one shouldn’t sit in the Sukkah.
- see Meshiv Tzedek by Rav Tzadok Hakohen of Lublin who wrote a complete pamphlet defending this practice of not sitting in the Sukkah.
- However, the Piskei Teshuvot concludes that one must be stringent based on the Sh”t Divrei Israel 1:200 that one shouldn't follow this minhag unless one has a tradition for it and it’s not comfortable outside.
- Aruch HaShulchan 668:3 writes that the gedolim have criticized those who just eat Kiddish in the Sukkah and then eat a meal inside. Mishna Brurah 668:5 also seems to hold that regarding eating one shouldn’t change from the ruling of S”A (which is based on an explicit gemara). Chacham Ovadia Yosef (Shu"t Yechave Daat 2:76 and Chazon Ovadyah (Sukkot pg 479)) also rules like Shulchan Aruc, as does Ben Ish Chai Vizot Haberacha: Halacha 13
- Darkei Moshe 668:2 writes that there’s room to be lenient regarding sleeping outside the Sukkah. Kitzur Shulchan Aruch 138:5 writes that while the minhag is to be lenient about sleeping in the sukkah, it's proper to be strict. Mishna Brurah 668:6 brings the opinion of the Gra who holds that one must sleep in the Sukkah on the eighth day but concludes that the minhag is to be lenient not to sleep in the Sukkah. However, the Bet Yosef 668 writes that sleeping is no different than eating in the Sukkah. Therefore, Chazon Ovadyah Sukkot (pg 479) and Ben Ish Chai Vizot Haberacha: Halacha 13 rule that Sephardim must sleep in the Sukkah like any other night of Sukkot.
- Nitai Gavriel (Hilchot Arbah Minim Sh”t 8), Sh”t Minchat Yitzchak 9:54, Rabbi Shlomo Zalman Auerbach in Sh”t Minchat Shlomo 1:19, Rabbi Ovadyah Yosef in Chazon Ovadyah Sukkot (pg 480-1) and Shu"t Yechave Daat 2:76, Ot Hei LeOlam (vol 2 pg 87c; quoted by Sdei Chemed Sukkah 1)
- Chacham Ovadia Yosef (Chazon Ovadyah Sukkot (pg 480-1) and Shu"t Yechave Daat 2:76) writes explicitly that if it’s not difficult to eat in the Sukkah such as one is staying by himself then he should sit in the Sukkah. However, Sh”t Minchat Shlomo 1:19 exempts such a person in all circumstances. So writes the Piskei Teshuvot 668:2 in name of Sh”t Minchat Yitzchak and Sh”t Minchat Shlomo.
- Shulchan Aruch 666:1, nitei Gavriel (Sukkot 81:4)
- S”A 666:1 writes that in Israel, if one must eat or sleep in a Sukkah on Shemini Aseret as there’s no room inside, then one should remove 4x4 Tefachim of S'chach to show that one doesn’t intend to add to the mitzvah (Baal Tosif). Mishna Brurah 666:5 explains that there’s no real Bal Tosif since one doesn’t violate Bal Tosif unless one intends to add to the mitzvah while it isn't the time of the mitzvah, however, there is an issue of appearing like adding to the mitzvah which is only applicable on Shemini Aseret. Nitei Gavriel 81:4 writes simply that same needs to be done if one needs to sleep in the Sukkah on Shemini Aseret. see also Simchas Torah-How to 'kosher' a Sukkah for Simchas Torah by Rabbi Zvi Sobolofsky
- Piskei Teshuvot 666:1 quotes the Bikurei Yacov 666:5 who says that using 4x4 invalid S'chach isn't sufficient against the Bigdei Yesha who permits but then adds that even the Bikurei Yacov agrees that putting a tarp 4x4 Tefachim on top or below the shach within 3 Tefachim of the S'chach is sufficient.
- Rama 666:1 writes that it’s clear that one isn’t sitting in the Sukkah to fulfill the mitzvah after Sukkot and so there’s no issue of Bal Tosif, however it would be an issue on Shemini Aseret because it’s so close to the holiday.
- Chazon Ovadyah pg 481 holds that’s there no issue of Bal Tosif before the mitzvah was done. Interestingly, Piskei Teshuvot 666:1 raises this issue (writes that it’s strange that no one addresses this issue) and concludes leniently and adds that it’s preferable to eat a meal or snack (KeBaytzah of mezonot) outside the Sukkah after eating in the Sukkah to show that one clearly doesn’t want to add to the mitzvah.
- Chazon Ovadia Sukkot p 451 quoting Masechet Sofrim
- Chazon Ovadia Sukkot 451-454
- Magen Avraham 668:3 quotes the Maharshal who writes that one shouldn't make Kiddish on Shemini Aseret until it is nightfall so as not to enter into a situation where there is a doubt if one will be able to make the bracha of Leshev BeSukkah. Many achronim including the Levush 668:1, Chaye Adam 153:5, Kitzur Shulchan Aruch 138:4, Aruch HaShulchan 668:6, Kaf HaChaim 668:10, Mishna Brurah 668:7, Nitei Gavriel 85:4, and Chazon Ovadyah (p. 669) agree. Chazon Ovadyah writes that there is even more reason for this law in Israel. [The term that the Magen Avraham uses, "Ad SheTechshach" in context clearly means certain nighttime which is after Tzet HaKochavim. This term is also used in the Mishna (Pesachim 99b) and there the Mishna Brurah 472:5 explains it to mean Tzet HaCochavim.]
- Kaf HaChaim 668:10 quoting Ben Ish Chai (Vezot HaBrachot #13), Nitei Gavriel 92:2