"Valentina Tereshkova, the first woman in space, orbited the Earth forty times before landed on June 19, 1963"
=> Valentina Tereshkova, ... before she landed on June 19, 1963.
=> Valentina Tereshkova, ... before landing on June 19, 1963.
These two are the only ways to express the idea.
Not everything can be omitted with equal ease. For example:
1. I took a shower before went out: BAD!
2. I took a shower before I went out. ("before" is a conjunctive)
3. I took a shower before going out. ("before" is a preposition)
There seems to be no logical explanation why #1 is bad. It is just the way English has developed.
No English speaker will say it like that, so you just need to get familiar with the accepted ways.
Not allowing the subject to be omitted is not a universal rule, though.
For phrases connected with "and / or" it often happens.
- I'll do this last thing and wrap up (what I'm doing".
- He made a big contribution and is now remembered for it.
- He either misunderstood it or is willfully distorting the fact.