It has not shown to be able to cast any genjutsu. And besides the sharigans few exceptions (ameratsu and susasanoo etc) each doujutsu is focused on one area. Byakugan taijustu, sharigan genjutsu, rinnegan ninjutsu
What if someone with EMS used Tsukuyomi on someone with Rinnegan, can the rinnegan do anything? Seeing as genjutsu is probably the quickest and easiest way to end fights, why did Nagato not use it? Maybe he could not?
Nagato isnt genjutsu type. Plus maybe he never learned any.
In theory rinnegan should be able to cast genjutsu. Perhaps not via eyes but in general any other should be within its reach
Braking it depends on the user not so much on rinnegan
The only genjutsu he's known to use is the ones he's placed in his subordinates minds, mental barrier techniques, as seen when Inoichi is trying to extract information from the ame genin but is confronted by blocks....
Thinking back on it, in the anime when fighting kakashi, it looks like he used genjutsu to dodge kakashi's raikiri.
On a note:
There was a discussion in the past wether some1 who hasn't activated rinnegan himself (first sharingan>Ms>EMS>rinnegan) cannot use the sharingan's genjutsu. Because you'll have to swap from rinnegan to EMS. I do not see why he cannot use other forms of genjutsu.
So if Nagato were to fight people like Itachi, Sasuke or Madara(all of which are skilled in genjutsu, agreed?) he wouldn't stand a chance seeing as they can just stop him before he moves?
We do not know yet if rinnegan is enough to overcome Tsukuyomi. But talking about 'normal' genjutsu i think pain would be able to recover from it quite fast. As the skill of the shinobi is key to breaking out of it.