wow... to be honest i can't see how this helps rinhwa at all, like she's only doing it to get back to yohwa right... like i thought yohwa was being a man to help their family not for any personal prestige? unless i missed something TT but if that's the case does rinhwa not think ahead...
It's mostly to do it to get back at Yohwa. She's so insecure and petty that she can't stand Yohwha getting things that she believes rightfully belongs to her. After her, she's the one who deluded herself into thinking the Emperor's actually in love with her.
I guess, in a way, she's hoping that by ruining Yohwa's reputation that either the 13th prince or the emperor will look upon her more favourably.
In summary, she sucks. She's just petty and dumb
Edit: Yonhwa technically drew first blood by getting Seo and the 9th princess together, knowing fully well that Rinwha still liked him at the time. However, that doesn't justify Rin revealing a secret that could bring harm to her family.