On happy occasions, Tachanun, the extra supplications added after the Shacharis and Minchah Amidah, is omitted. The Shulchan Arukh (Orach Chaim 131:4,6-7) lists such days. However, some congregations stretch the definition “happy occasions” to the limit, and sometimes beyond. What do you do if you are in a synagogue where the congregation improperly omits Tachanun?
Rav Natronai Gaon (quoted in Tur, Orach Chaim 131) states that Tachanun is only optional. The Arukh Ha-Shulchan (Orach Chaim 131:2) states that in the approximately thousand years since Rav Natronai’s time, Jews have universally accepted Tachanun as an obligation. This is, presumably, similar to the Talmudic statement (Berakhos 27b) that Ma’ariv is optional. The Rif (Berakhos 19a) states that Ma’ariv was subsequently established as an obligation (although Tosafos [Berakhos 26a sv. ta’ah] says it was always obligatory). Just like we treat Ma’ariv as a requirement, we should also treat Tachanun as a requirement. Therefore, you cannot simply omit Tachanun without proper justification.
R. Asher Bush (Sho’el Bi-Shlomo, ch. 6) points out that the Rambam (Mishneh Torah, Hilkhos Tefillah 7:17; 9:5,8) includes Tachanun in the order of prayers for both an individual and a congregation. Evidently, Tachanun is not merely a congregational passage but part of the daily prayer services that must be recited in private or public. The Shulchan Arukh does not indicate any difference between an individual’s and a congregation’s prayer text in this respect. Even if the congregation skips Tachanun, you still have to say it.
However, R. Bush states that even in such a case, you should not be obvious about saying Tachanun. While the custom is to lean forward during the beginning of Tachanun (see this post: link), in order to avoid conflict you should not do so in such a situation.