Khi \(\sin A = \frac{{\cos B + \cos C}}{{\sin B + \sin C}}\) thì tam giác ABC là tam giác gì?

\(\sin A = \frac{{\cos B + \cos C}}{{\sin B + \sin C}} = \frac{{2\cos \frac{{B + C}}{2}\cos \frac{{B - C}}{2}}}{{2\sin \frac{{B + C}}{2}\cos \frac{{B - C}}{2}}} = \frac{{\cos \frac{{B + C}}{2}}}{{\sin \frac{{B + C}}{2}}} = \frac{{\cos \left( {\frac{\pi }{2} - \frac{A}{2}} \right)}}{{\sin \left( {\frac{\pi }{2} - \frac{A}{2}} \right)}} = \frac{{\sin \frac{A}{2}}}{{\cos \frac{A}{2}}}\)

Suy ra: \(\sin A = \frac{{\sin \frac{A}{2}}}{{\cos \frac{A}{2}}}\)

⇒ \(2\sin \frac{A}{2}\cos \frac{A}{2} = \frac{{\sin \frac{A}{2}}}{{\cos \frac{A}{2}}}\)

⇒ \(2{\cos ^2}\frac{A}{2} = 1 \Rightarrow \cos A = 0 \Rightarrow A = 90^\circ \)

Suy ra tam giác ABC vuông tại A.