Other answers mention that <PERSON> is arrogant and not that smart, which it true, but there's more to why he made his decision.
The quote mentioned in the question
To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret.
is from after <PERSON> helped <PERSON> and <PERSON> was killed as a result. First, at that point, it was a bit late for <PERSON> to backtrack and be like "Oh so you don't even know for sure how to do it? Then I'm out." and go back to the Jedi. Second, keep in mind that this was the first time <PERSON> mentioned the ability to cheat death with much less degree of confidence than before.
At first, he only hinted that it's possible
— Darth <PERSON> was a dark lord of the Sith so powerful and so wise, he could use the Force to influence the midi-chlorians to create life. He had such a knowledge of the dark side, he could even keep the ones he cared about from dying.
...
— Is it possible to learn this power?
— Not from a Jedi.
The next time they met, his statement was very confident.
Learn to know the dark side of the Force, and you will be able to save your wife from certain death.
As it was when <PERSON> was about to kill <PERSON>.
I have the power to save the one you love.
It means that <PERSON> thought that <PERSON> already knew how to cheat death before making his choice. But saving <PERSON> alone wasn't enough for him to betray the Jedi, even though he already had some doubts about them when the council asked him to spy on <PERSON>.
You're asking me to do something against the Jedi code, against the Republic, against a mentor and a friend.
Even after that he still told <PERSON> that <PERSON> is a Sith as soon as he found it out, which almost certainly meant that <PERSON> won't be around to help him save <PERSON>. 605
That was also the reason why he simply didn't have a chance to question <PERSON> about his ability to cheat death. He couldn't accept him being a Sith back then. Next time they met, he obviously didn't have the time to ask questions.
Still <PERSON> wasn't sure if he's making the right choice, as <PERSON> noted
For your own good, stay out of this affair. I sense a great deal of confusion in you, young <PERSON>. There is much fear that clouds your judgement.
Only when <PERSON> was going to kill an "unarmed" and "weak" old man, which was definitely against the Jedi teachings, he made the final choice. At that point, his disappointment in the Jedi was almost as good of a reason for him to support <PERSON> as saving <PERSON>. 605