Cho $A, B, C$ và $a, b, c$ lần lượt là các góc và các cạnh của tam giác $ABC.$ Chứng minh rằng:
                                 $\frac{sin (A - B)}{sin C} = \frac{a^2 - b^2}{c^2}$
$\begin{array}{l}
\,\,\frac{{{a^2} - {b^2}}}{{{c^2}}} = \frac{{\sin^ 2A - \sin^ 2B}}{{\sin^ 2C}} = \frac{{(1 - \cos 2A) -
(1 - \cos 2B)}}{{2{{\sin }^2}C}}\\
 = \frac{{\cos 2B - \cos 2A}}{{2{{\sin }^2}C}} = \frac{{\sin (A + B)\sin ( A-B)}}{{{{\sin }^2}C}}\\
 = \frac{{\sin C\sin (A - B)}}{{{{\sin }^2}C}} = \frac{{\sin (A - B)}}{{\sin C}}\,\,(dpcm)
\end{array}$

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