As anyone noticed that Sesshou has actually not fought Inuyasha for a long time. I mean there are several battles where he could have really hurt Inu or killed him, but in the end decided not too.
Like when Inu lost control of his demon blood, and he came to "Investigate it". He could have killed Inuyasha there no problem, but instead choose to simply knock him out. If I remember correctly Kagome asked him why he didn't kill him, and he actually had to think about his answer for a couple of senconds, then choose to say something like
" What would be the point in killing him since he is not aware of me doing it. "
I remerber thinking....." Sure Sesshou that's your excuse. "... Apparently Kagome didn't buy it either, cause I remerber her thinking something about Sesshou loving his brother more than he let on.
I think the only battle where Sesshou truly wanted to kill Inu was the first one in their father's tomb. Maybe after that battle Sesshou realised that Inu was strong after all. Not that he'd ever andmit it ofcourse. XD
But after that it was more like...Since i'm your big brother...I have to show you how to work Tetsaiga. Since Inuyasha would have never accepted Sesshoumaru teaching him how to use Tetsaiga, and let's face it, I really don't see Sesshoumaru actually walking up to Inuyasha and telling him that. Fighting him was the only way to make Inuyasha learn to use Tetsaiga. Well at least it worked.