You're on the right lines. cos = adj/hyp and sin = opp/hyp, so cos² = adj²/hyp² and sin² = opp²/hyp². So cos² + sin ² = (adj² + opp²)/hyp².

But Pythagoras's Theorem says that a² + b² = c², where c is the hypotenuse, so that means that adj² + opp² = hyp². Therefore, cos² + sin² = hyp²/hyp² = 1.

That's an identity that you'll be using a lot while you're studying trig, so remember it well.