According to the book, Tang Soo Do History by Dan Segarra:
https://ia802901.us.archive.org/3/items/TangSooDoHISTORYV15b_201808/TangSooDo_HISTORY_V15b.pdf
"In 1936 Master Yang Kuk Jin (楊鞠振/楊麴震) was a well-known master of northern Kung Fu particularly Tai Chi and Tan Tui... Master Yang trained them in Bo Bup (stepping methods), Ryun Bup (conditioning methods), Dam Toi Ship Ee Ro (twelve longfist forms), and Tae Kuk Kwon (Tai Chi)."
Back to me...
So for those of you who don't know, northern shaolin long-fist (Chang Chuan) kung-fu is a kicking oriented, long range style. The reference to Tan Tui means "springing legs", which is a kind of sub-style of long-fist kung-fu.
Tan Tui may be a complete style in and of itself (or a number of styles), but it's usually known as a collection of drills (like miniature forms) typically taught as basic exercises prior to learning the main long-fist forms.
I practiced the Tan-Tui drills as part of my long-fist training. It's quite common to see as part of long-fist training. It was commonly taught in the Jing Wu (Chin Woo) schools (sort of like a Chinese version of the YMCA) and is probably still taught in their basic curriculum.
Whether Yang, Kuk Jin taught Yang style taiji (Tai Chi) or not is unknown. I'm not sure anyone knows the lineage of the styles he taught. My guess is that it would be Yang style, since it's very common to see it taught along with Long-Fist kung-fu in the same school. Yang style taiji is the most popular form of taiji as well.
Incidentally, Yang is a very common Chinese name. There's probably no link between Yang, Kuk Jin and the founders of Yang style Taiji.
It's hard to see how any kung-fu or taiji made it into Tangsoodo. But maybe there are people out there who have studied that question and can give you a definitive answer.
Hope that helps.