Chứng minh rằng trong mọi tam giác ABC ta đều có

sin A + sin B + sin C = \(4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}\).

Lời giải

\(VT = \sin A + \sin B + \sin C\)\( = 2\sin \frac{{A + B}}{2}\cos \frac{{A - B}}{2} + 2\sin \frac{C}{2}\cos \frac{C}{2}\).

Mặt khác, trong tam giác ABC, ta có A + B + C = π nên \(\frac{{A + B}}{2} = \frac{\pi }{2} - \frac{C}{2}\).

Từ đó suy ra: \(\sin \frac{{A + B}}{2} = \cos \frac{C}{2},\,\sin \frac{C}{2} = \cos \frac{{A + B}}{2}\).

Vậy \(VT = 2\sin \frac{{A + B}}{2}\cos \frac{{A - B}}{2} + 2\sin \frac{C}{2}\cos \frac{C}{2}\)

\( = 2\cos \frac{C}{2}\cos \frac{{A - B}}{2} + 2\cos \frac{{A + B}}{2}\cos \frac{C}{2}\)

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

\( = 2\cos \frac{C}{2}.2\cos \frac{{\frac{{A - B}}{2} + \frac{{A + B}}{2}}}{2}\cos \frac{{\frac{{A - B}}{2} - \frac{{A + B}}{2}}}{2}\)

\( = 4\cos \frac{C}{2}\cos \frac{A}{2}\cos \left( { - \frac{B}{2}} \right)\)

\( = 4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2} = VP\) (điều phải chứng minh).