Analisando os ângulos:\begin{matrix} \alpha + \beta + \hat{C} = 180º &\Rightarrow& \hat{C} = 180º - (\alpha + \beta ) &∴& \sin{\hat{C}} = \sin{(\alpha + \beta)} \end{matrix}Lei dos senos:\begin{matrix} { \dfrac{\overline{AC}}{\sin{\beta}} = \dfrac{\overline{BC}}{\sin{\alpha}} } &\Rightarrow& \overline{BC} = 2\cdot {\dfrac{\sin{\alpha}}{\sin{\beta}}} \end{matrix}Área do triângulo: \begin{matrix} A &=& { \dfrac{\overline{AC} \cdot \overline{BC} \cdot \sin{\hat{C}}}{2} } &=& 2\cdot (\sin{\alpha}\cos{\beta} + \sin{\beta}\cos{\alpha}) \cdot {\dfrac{\sin{\alpha}}{\sin{\beta}}} \end{matrix}\begin{matrix} A = 2\sin{\alpha}^2\cot{\beta} + 2\sin{\alpha}\cos{\alpha} \\ \\ \color{gray}{\sin{2\alpha} =2\sin{\alpha}\cos{\alpha} } \\ \\ \fbox{$A = 2\sin{\alpha}^2\cot{\beta} + 2\sin{2\alpha}$} \\ \\ Letra \ (A) \end{matrix}