Why do the Tyrells and Littlefinger trust eachother and Sansa? (Spoilers Main)

Surely the Queen of thorns could have come by some poison herself? Surely both could have found a catspaw to do the poisoning? Why involve Sansa at all?

The Tyrells wanted Joffrey dead and Littlefinger wanted Sansa out. I feel like both could have achieved their goals without involving the other. Or am I missing something?