Proving that matrices in $O(2)$ are of one of two forms
Solution 1:
Since $a^2+b^2=1$, there is some $\theta\in\Bbb R$ such that $a=\cos\theta$ and $b=\sin\theta$. And, since $(c,d)$ is orthogonal to $(a,b)$,$$(c,d)=\lambda(-b,a)=(-\lambda\sin\theta,\lambda\cos\theta)$$for some $\lambda\in\Bbb R$. But$$\pm1=\begin{vmatrix}\cos\theta&-\lambda\sin\theta\\\sin\theta&\lambda\cos\theta\end{vmatrix}=\lambda;$$so, $\lambda=\pm1$ and you're done.