Aside from a bit of hesitant uncertainty shortly after the war, Kakashi quickly came to wholly trust Sasuke. Although I'd say Kakashi's trust in Sasuke is less of a product of his actual relationship to Sasuke and more a product of Kakashi's relationship to Naruto. Sasuke isn't the most trustworthy, as he once before had betrayed his teammates and everyone in Konoha. But Kakashi trusts Naruto, and since Naruto trusts Sasuke, Kakashi trusts Sasuke too. In fact, the shinobi nations' trust in Sasuke comes completely from Naruto. People don't trust Sasuke, they trust Naruto so much that they're willing to trust a former criminal because of that trust, until of course, eventually Sasuke proves himself trustworthy on his own. 

As we see in the Book of Sunrise arc, Kakashi has no problem with requesting Sasuke's help when it's needed. Asking for Sasuke's help with missions is pretty much the only substantial interaction Kakashi and Sasuke have post-war. After all, because of his great abilities and skill, Sasuke is a valuable asset and a big help.

I don't acknowledge Sakura Hiden as canon because of how much I disliked the novel, but I'll mention it anyway for those who do. Towards the end of Sakura Hiden, Sakura is kidnapped by Kido. When Sasuke learns of Sakura's capture he rushes back to save her, but Sakura escaped by herself before Sasuke gets there, so Sasuke dealt with the enemies who escaped from Sakura's battle with Kido before he slipped quietly away without visiting any of his teammates.

Kakashi is the one who reveals to Sakura, Naruto, Sai, and Ino that Sasuke had returned to help save them. Kakashi seems delighted by the Uchiha sweeping in to help save the day, since Kakashi is described as "grinning playfully" when he hints to Team 7 who dispatched the remainder of Kido's forces, additionally laughing later on in the conversation when he comments that he is certain that it was the "real" Sasuke who saved them, not Kido's impersonator who had been framing Sasuke for his crimes. So in that light novel, Kakashi was pleased and delighted by Sasuke's brief return, cheerfully and playfully relaying news to Team 7.

In the anime Book of Sunrise arc, Sasuke requests permission to investigate the bombing incident independently. Upon receiving that written request, Kakashi muses on how long Sasuke's journey will continue. This scene is a little tricky to analyze precisely, because to me, Kakashi's reaction comes off differently in the sub version than in the English dub version, so I'll analyze both separately.

Sub Version

In the subbed version, after reading Sasuke's letter of request to investigate the case on his own, Kakashi tilts his head back and muses, "Just how long will your journey continue?"

From the tone of his voice, I get a sense of melancholy exasperation. It sounds like Kakashi wants Sasuke to return home, but even when Kakashi orders Sasuke to return, Sasuke chooses to work independently and keep his distance from Konoha.

His voice's inflection evokes a sense of wary exasperation with Sasuke as Kakashi wonders when Sasuke will stop going on with his self-imposed exile and return home to Konoha already. It makes one think that Kakashi is a bit irked at Sasuke for continuing his journey with no sign of stopping, even while Naruto and Sakura are still awaiting their friend's return.

This indicates that Sasuke's distance bothers Kakashi some, and he wishes Sasuke would just return to Konoha instead of working nonstop so far away from the Leaf and his friends.

I suspect Kakashi's exasperation is out of regard for Naruto and Sakura or to wonder what the Uchiha is going to do with his life, perhaps wondering with a bit of exasperation if the Uchiha is ever going to settle down and get a life.

Dub Version

On the other hand, Kakashi's dub version evokes quite a different impression via the changes in the tonal inflictions of Kakashi's English voice and the slight changes in dialogue.

This time Kakashi's line is, "Just how long will your journey be forced to continue?"

It's only a three-word difference, but it changes the meaning quite a bit. In the sub version, Kakashi seemed exasperated by Sasuke's self-appointed quest to travel outside the village for so long, leaving Naruto and Sakura waiting for Sasuke to come back.

But in the dub version, the phase "be forced" implies a loss of agency that the sub version doesn't have. In the sub, Kakashi seems like he's holding Sasuke responsible for his extended absence. In the dub, Kakashi seems more like he's blaming external circumstances as keeping Sasuke away from Konoha and preventing him from being able to return home.

Perhaps it's a reflection of Kakashi's experience as Hokage, as Kakashi is quickly learning there is always more work to be done, perhaps he feels sorry for Sasuke as he imagines Sasuke finding himself in the same boat Kakashi is now in: there's always more work to be done, more missions to complete, more crises to be resolved. In the dub, Kakashi's exasperation isn't targeted on Sasuke's own stubborn refusal to return to the Leaf Village, but aimed towards unfortunate external events that prevent Sasuke from having time to return to Konoha.

Additionally, Kakashi's subbed voice sounds more somber and melancholy, whereas Kakashi's dubbed voice seems less tragic, merely sounding curious in a mild-mannered way.

Sasuke's request to investigate the mission on his own is polite, showing that he is respectful to authority in a way he wasn't before. He's taking his parole seriously, knowing that he still hasn't fully earned everyone's trust and doesn't deserve to be walking free. So Sasuke is polite and respectful as he requests permission to investigate independently instead of returning to Konoha as Kakashi requested, using phrasing such as "I respectfully ask for your understanding" (dub version) and "please approve" (sub version). Sasuke had fallen back on decorum and is respectful to authority figures now (at least he can be when he chooses to); though it is formal and impersonal.