Now we just saw the chapter 603 where obito fuses with zetsu to save rin.

But its already stated in the manga that obito is angry that kakashi let rin die

Its stated in the manga that Mangekyō Sharingan is acquired through the trauma experienced from the death of the person closest to the Sharingan user. So my theory is that because obito will definitely watch rin die in front of him(in the next chapter) and because Obito loves Rin to death this will traumatize obito causing him to gain the Mangekyo sharingan.