Cho tam giác ABC. Với tân/2, tanB/2,tanC/2 lập thành cấp số cộng nếu và chỉ nếu

* Hướng dẫn giải

$\tan \left(\frac{A}{2}\right)$, $\tan \left(\frac{B}{2}\right)$ , $\tan \left(\frac{C}{2}\right)$ lập thành cấp số cộng

2$\tan \left(\frac{B}{2}\right)$ = $\tan \left(\frac{A}{2}\right)$ + $\tan \left(\frac{C}{2}\right)$ 

$$\begin{gathered} \iff 2\frac{\sin \left({\displaystyle \frac{B}{2}}\right)}{\cos \left({\displaystyle \frac{B}{2}}\right)}=\frac{\sin \left({\displaystyle \frac{A}{2}}\right)}{\cos \left({\displaystyle \frac{A}{2}}\right)}+\frac{\sin \left({\displaystyle \frac{C}{2}}\right)}{\cos \left({\displaystyle \frac{C}{2}}\right)} \\ \Rightarrow 2\frac{\sin \left({\displaystyle \frac{B}{2}}\right)}{\cos \left({\displaystyle \frac{B}{2}}\right)}=\frac{\cos \left({\displaystyle \frac{B}{2}}\right)}{\cos \left({\displaystyle \frac{A}{2}}\right).\cos \left({\displaystyle \frac{C}{2}}\right)} \end{gathered}$$

$$\begin{gathered} \sin \left(\frac{B}{2}\right)\left(\cos \left(\frac{A+C}{2}\right)+\cos \left(\frac{A-C}{2}\right)\right) \\ ={\cos }^2\left(\frac{B}{2}\right) \\ \iff \sin \left(\frac{B}{2}\right)\left(\sin \left(\frac{B}{2}\right)+\cos \left(\frac{A-C}{2}\right)\right) \\ -{\sin }^2\left(\frac{B}{2}\right) \\ ={\cos }^2\left(\frac{B}{2}\right)-{\sin }^2\left(\frac{B}{2}\right) \end{gathered}$$

$$\begin{gathered} \iff \sin \left(\frac{B+A-C}{2}\right)+\sin \left(\frac{B-A+C}{2}\right) \\ =2\cos \left(B\right) \\ \iff \cos \left(C\right)+\cos \left(A\right)=2\cos \left(B\right) \end{gathered}$$

Đáp án cần chọn là C