As part of a proof that all 2x2 orthogonal matrices either represent rotations or reflections (Introduction to Applied Linear Algebra, exercise 10.37), I came across the below.
Let $a, b, c, d \in \mathbb R$. Given that $a^2 + c^2 = 1$, $b^2 + d^2 = 1$, and $ab + cd = 0$, I need to show that $|ad - bc| = 1$.
I've tried many different approaches here, but can't seem to figure out what the trick is. Any hint would be much appreciated.
Also, is there a standard approach for solving problems of this kind? If so, what is it?