Giải thích các bước giải:
Ta có:
\(\begin{array}{l}\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}} = 2R\\
\Rightarrow \frac{{\sin A}}{{\sin C}} = \frac{a}{c};\,\,\,\,\,\,\frac{{\sin B}}{{\sin C}} = \frac{b}{c}\\
\frac{{\sin A.\cos B + \sin B.{\mathop{\rm cosA}\nolimits} }}{{{\mathop{\rm sinC}\nolimits} }}\\
= \frac{{\sin A}}{{\sin C}}.\cos B + \frac{{\sin B}}{{\sin C}}.{\mathop{\rm cosA}\nolimits} \\
= \frac{a}{c}.\frac{{{a^2} + {c^2} - {b^2}}}{{2ac}} + \frac{b}{c}.\frac{{{c^2} + {b^2} - {a^2}}}{{2cb}}\\
= \frac{{{a^2} + {c^2} - {b^2}}}{{2{c^2}}} + \frac{{{c^2} + {b^2} - {a^2}}}{{2{c^2}}}\\
= \frac{{{a^2} + {c^2} - {b^2} + {c^2} + {b^2} - {a^2}}}{{2{c^2}}}\\
= \frac{{2{c^2}}}{{2{c^2}}}\\
= 1\\
\Rightarrow \sin A\cos B + \sin B\cos A = \sin C
\end{array}\)
