Because of the Sun, which is 400 times larger than the Moon

Although looking innocuously harmless, these questions are deep and tricky. Let us take the first one.

1. Because of the Sun, which is 400 times larger than the Moon, But also 400 times farther away from Earth, so both the Sun and the Moon have the same apparent size in the sky. ----- This is a complex sentence with a sub - clause and a main clause. The best tool to connect a sub clause with the main clause is the comma; a coordinating conjunction can be used only when connecting two ICs. Therefore, the first sentence is grammatically wrong.

2. The Sun, which is 4oo times larger than the Moon, is also 4oo times farther away from Earth, so the sun and the Moon have the same apparent size in the sky. ------ This is a tougher sentence to comment. Apparently, there is nothing grammatically wrong in this sentence and the word so, a coordinating conjunction is rightly used to connect 2 ICs in a compound sentence as this. However, it is only the usage that is a problem. There is a custom in formal English writing that the word so is not used to indicate either a purpose of effect.

For example, it is not correct to say, -----I prepared hard for my GMAT, and so I could cross 700. ----Here, you are doing something with a purpose in mind. That purpose cannot be conjugated by a so. You must say, I prepared hard for my GMAT in order to or simply to cross 700

Similarly, it is not correct to use so to connect a result or an effect.
Ex: I prepared hard for GMAT so I crossed 700 – this is wrong because you are trying to tie the result of a cause in the second part. It is proper to say that I prepared hard for GMAT and therefore I crossed 700.

The second sentence falls in this category. The sun and the moon appearing of about the same size is the result of some factor. It is not right to connect that result with a so. You must say,

The Sun, which is 4oo times larger than the Moon, is also 4oo times farther away from Earth, and therefore, the sun and the Moon have the same apparent size in the sky

So is often used to describe the extant of some thing. We say, he is so impressive and is selected in any interview first time. We can connect two ICS, which do not exhibit purpose of result by a so, as in ---The child was unbearably heavy, so we couldn’t physically carry him.--

Some sentences are introduced with so, when we need to emphasize something. Ex;

1. So innocent were the villagers that, the Insurance Agent got a policy from each one, by promising rains next month.
2. So adamant is my newly married son that he refuses to heed to any one other than his wife.