Cho tam giác MNP. Khẳng định nào trong các khẳng định sau là đúng?

* Hướng dẫn giải

Tổng ba góc trong 1 tam giác bằng 180$°$  nên:

$$\begin{gathered} \hat{M}+\hat{N}+\hat{P}=180°\iff \hat{M}+\hat{N}=180°-\hat{P} \\ \iff \frac{\hat{M}+\hat{N}}{2}=90°-\frac{\hat{P}}{2}\Rightarrow \cos \frac{M+N}{2}=\sin \frac{P}{2};\sin \frac{M+N}{2}=\cos \frac{P}{2} \\ \sin M+\sin N+\sin P=2\sin \frac{M+N}{2}.\cos \frac{M-N}{2}+2\sin \frac{P}{2}.\cos \frac{P}{2} \\ =2\cos \frac{P}{2}.\cos \frac{M-N}{2}+2\sin \frac{P}{2}.\cos \frac{P}{2}=2\cos \frac{P}{2}.\left(\cos \frac{M-N}{2}+\sin \frac{P}{2}\right) \\ =2\cos \frac{P}{2}.\left(\cos \frac{M-N}{2}+\cos \frac{M+N}{2}\right)=2\cos \frac{P}{2}.2\cos \frac{M}{2}.\cos \frac{N}{2} \\ =4\cos \frac{M}{2}.\cos \frac{N}{2}.\cos \frac{P}{2} \end{gathered}$$