We may think we already know the Pythagorean identity. Pythagoras was that old Greek dude who obsessed over triangles. A lot. But that's not the identity we're talking about. It's this one:
sin2 θ + cos2 θ = 1
One is the loneliest number, and so are sin2 and cos2, together. We're not sure how the math on that works out.
We can find out, though. Let's go back to our original definitions for sine and cosine. We'll save going back to the future for another time.
If we square them both and add them together, we get the start of a great party (because it's hip to be square).
And we've got it. That stuff up top? That's a2 + b2. Down bottom is c2. Old man Pythagoras has something to say about those: they're equal. Because they are equal, the whole thing equals 1.
At least, that proves it for angles in right triangles. Pythagoras didn't have nearly as much to say about all the other triangles and angles out there. At this point, we're just going to wave our hands, tell you that it's actually true all the time, and ask you to trust us. We've earned that much trust, haven't we? Forget about the shaving cream incident; that's ancient history by now.