ITA 2013 — Questão 27

Encontre os pares $(\alpha, \beta) \in \,]0, \frac{\pi}{2}[\, \times \,]0, \frac{\pi}{2}[$ que satisfazem simultaneamente as equações

$(\operatorname{tg}\alpha + \operatorname{cotg}\beta)\cos\alpha\operatorname{sen}\beta - 2\cos^2(\alpha - \beta) = -1 \quad \text{e} \quad \sqrt{3}\operatorname{sen}(\alpha + \beta) + \cos(\alpha + \beta) = \sqrt{3}\,.$