Cho tam giác ABC chứng minh rằng:
\(\cos \widehat A + \cos \widehat B - \cos \widehat C = 4\cos \frac{{\widehat A}}{2}.\cos \frac{{\widehat B}}{2}.\sin \frac{{\widehat C}}{2} - 1\)
\(VT = \cos \widehat A + \cos \widehat B - \cos \widehat C\)
\( = 2\cos \frac{{\widehat A + \widehat B}}{2}.\cos \frac{{\widehat A - \widehat B}}{2} + 2{\sin ^2}\frac{{\widehat C}}{2} - 1\)
\( = 2\sin \frac{{\widehat C}}{2}.\cos \frac{{\widehat A - \widehat B}}{2} + 2{\sin ^2}\frac{{\widehat C}}{2} - 1\)
\( = 2\sin \frac{{\widehat C}}{2}\left( {\cos \frac{{\widehat A - \widehat B}}{2} + \sin \frac{{\widehat C}}{2}} \right) - 1\)
\( = 2\sin \frac{{\widehat C}}{2}\left( {\cos \frac{{\widehat A - \widehat B}}{2} + \cos \frac{{\widehat A + \widehat B}}{2}} \right) - 1\)
\( = 4.\cos \frac{{\widehat A}}{2}.\cos \frac{{\widehat B}}{2}.\sin \frac{{\widehat C}}{2} - 1\).