When Kyo first appears, he comes off as being a very rude and careless individual. He acts without thinking. However, a few episodes later, we see that Kyo really isn't as heartless as he makes himself out to be. He regrets saying cruel things to Tohru and he often wonders if anyone could really like a guy such as himself. From this point, we begin to learn that Kyo has a serious figure underneath his reckless personality.
All of Kyo's grief stems from the fact that he's cursed with the spirit of the cat. No, not the little orange cat that we often see through the series... the true cat has a different form. It's not cute and cuddly; it's not even orange. It's a monster. A hideous, frightful monster.
...So you can see why Kyo doesn't want anyone outside of the Souma family to find out about his true form. He has been ridiculed his whole life, by friends and family alike. His own mother feared him. She would often tell Kyo that she loved him very much but she would never take him outside, claiming him to be too cute to share. She would always check his wrist to see if he had his bracelet on. She was afraid of a small child... she was afraid of her own son.
Aside from Kagura and Kazuma, Kyo has never experienced love or kindness. Everyone was simply too afraid to get near him, in fear of his curse. When Tohru came into Kyo's life, he experienced kindness and affection from someone outside of his family for the first time. With Tohru's help, Kyo learned that he really isn't a bad person... he too has good qualities about him that he never saw before. Therefore, Tohru's care is one of the most important things in the world to Kyo. He would do anything to protect that. Which is why he neglects to tell or show her about his true form.
In Kyo's mind, if Tohru learned that he was a freak, a monster, she would end up just like everyone else. She would be afraid of him and he would lose the only form of affection he has ever had (aside from Kagura and Kazuma). In the end, Tohru accepts him as he is and he no longer has a fear of losing her.
Once his fear of losing Tohru is gone, Kyo only has to deal with his raging temper. Kyo believes that it's Yuki's fault that he has been ridiculed all of his life. The mouse tricked the cat and because of the cat's vengeance, it turned into a monster. Kyo believes that if he can succeed at beating Yuki, he will be accepted as a true member of the Souma family, not some outcast.
It's important to understand Kyo's rivalry with Yuki, to better understand Kyo himself. In Kyo's mind, Yuki has it all. He hasn't had the torment of being shunned his whole life.. he's been accepted into all of the family events. He's a loved person. While this isn't necessarily true, Kyo doesn't know it. Yuki has had a horrible life because he's a part of the Souma family. Kyo has had a horrible life because he isn't a part of the Souma family. A distinct parallel exists between the two young men but neither of them know it.
Both Kyo and Yuki are too stubborn to sit down and listen to each other. Kyo doesn't believe that Yuki will understand what he has had to go through and because Yuki considers Kyo to be too uncivilized, he has no wish to listen to what he believes to be Kyo's dribble. Kyo is too determined to be loved by his family that he has become blind. He doesn't realize that the Souma family is not what he believes it to be. Aside from that, he doesn't know what will happen if he does beat Yuki.
Yes, Kyo's rivalry with Yuki can be considered him acting before really thinking. But Kyo has thought about this. If you have been shunned and feared all of your life, you would seek and end to it all. And for Kyo, his end is by beating Yuki, to prove that he is just as good as the mouse is, if not better.
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