Let $A, B \in \mathscr M_n (\mathbf R)$ be two orthogonal matrices such that $\det A + \det B = 0.$
Prove that $A+B$ is singular.
No words needed:
$$
\begin{align*}
\det (A + B) &= \frac{1}{2} (\det (A+B) + \det(A+B))
= \frac{1}{2} (\det A \det (\mathbf 1 + \trans A B) + \det B \det (\mathbf 1 + \trans B A))\\
&= \frac{1}{2} (\det A \det (\mathbf 1 + \trans B A) + \det B \det (\mathbf 1 + \trans B A))
= \frac{1}{2} (\det A + \det B) \det (\mathbf 1 + \trans B A) = 0.
\end{align*}
$$