Thu gọn A = sin^2 alpha + sin^2 beta + 2sin alpha.sin beta.cos(alpha + beta

* Hướng dẫn giải

Đáp án A

$$\begin{gathered} A={\sin }^2\alpha +{\sin }^2\beta +2\sin \alpha \sin \beta .cos\left(\alpha +\beta \right) \\ ={\sin }^2\alpha +{\sin }^2\beta +2\sin \alpha \sin \beta .\left(cos\alpha .cos\beta -\sin \alpha \sin \beta \right) \\ ={\sin }^2\alpha +{\sin }^2\beta -2{\sin }^2\alpha {\sin }^2\beta +2\sin \alpha \sin \beta cos\alpha .cos\beta \\ ={\sin }^2\alpha (1-{\sin }^2\beta )+{\sin }^2\beta (1-{\sin }^2\alpha )+2\sin \alpha \sin \beta cos\alpha .cos\beta \\ ={\sin }^2\alpha co{s}^2\beta +{\sin }^2\beta co{s}^2\alpha +2\sin \alpha \sin \beta cos\alpha .cos\beta \\ ={(\sin \alpha cos\beta +\sin \beta cos\alpha )}^2={\sin }^2\left(\alpha +\beta \right) \end{gathered}$$