11.1 Given a triangle, where two angles $\alpha, \beta$ satisfy the relation $\cos \alpha+\cos \beta=\sin \alpha+\sin \beta$. Is this triangle necessarily a right triangle?
Answer: necessarily. Solution. Rewrite the given relation as $2 \cos (\alpha+\beta) / 2) \cos ((\alpha-\beta) / 2)=2 \sin ((\alpha+\beta) / 2) \cos ((\alpha-\beta) / 2)$.

Notice that $\cos ((\alpha-\beta) / 2) \neq 0$, otherwise $\alpha-\beta=180^{\circ}$. Then $\cos (\alpha+\beta) / 2)=\sin ((\alpha+\beta) / 2)$, and therefore $\operatorname{tg}((\alpha+\beta) / 2)=1$ (here division by $\cos (\alpha+\beta) / 2)$ is possible, since the sine and cosine of the same angle cannot be 0 simultaneously). Thus, $(\alpha+\beta) / 2=45^{\circ}$, i.e., $\alpha+\beta=90^{\circ}$.