Biết cos alpha + cos beta = m; sin alpha + sin beta = n. Tính cos(alpha − beta

* Hướng dẫn giải

Đáp án A

$$\begin{gathered} \cos \alpha + \cos \beta = m; \sin \alpha + \sin \beta = n \\ \Rightarrow m^2+n^2={(cos\alpha +cos\beta )}^2\text{+(}sin\alpha +sin\beta {)}^2 \\ =co{s}^2\alpha +2cos\alpha cos\beta +co{s}^2\beta \text{+}si{n}^2\alpha +2sin\alpha sin\beta +si{n}^2\beta \\ =\left(co{s}^2\alpha \text{+}si{n}^2\alpha \right)+(co{s}^2\beta +si{n}^2\beta )++2(cos\alpha cos\beta +sin\alpha sin\beta ) \\ =1+1+2cos\left(\alpha -\beta \right)=2+2cos\left(\alpha -\beta \right) \end{gathered}$$

Do đó $cos\left(\alpha -\beta \right)=\frac{m^2+n^2-2}{2}$