If one holds that Jews living outside of Israel should sit in a sukkah because of Sfeika Dyoma (We aren't sure what day it is), then why shouldn't we shake lulav and esrog? Worst comes to worst we are ...
In the diaspora, where two days of yomim tovim are celebrated, the second day is typically an exact repetition of the first day. We say the same prayers, make a second Seder on Pesach, make kiddush ...
Shouldn't the obligation to sit in the Sukkah also fall into the category of Sefeika D'yoma? If so why do some not eat in the Sukkah on Shemini Atzeres?
Why do we start saying Mashiv HaRuach on Shemini Atzeres yet we do not say V'Sain Tal Umatar until Dec 4 / Dec 5 (In Chutz L'Aretz)? Why should they not be on the same day when they are similar ...