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$\begin{array}{l}
\,\,\frac{{{a^2} - {b^2}}}{{{c^2}}} = \frac{{\sin^ 2A - \sin^ 2B}}{{\sin^ 2C}} = \frac{{(1 - \cos 2A) -
(1 - \cos 2B)}}{{2{{\sin }^2}C}}\\
= \frac{{\cos 2B - \cos 2A}}{{2{{\sin }^2}C}} = \frac{{\sin (A + B)\sin ( A-B)}}{{{{\sin }^2}C}}\\
= \frac{{\sin C\sin (A - B)}}{{{{\sin }^2}C}} = \frac{{\sin (A - B)}}{{\sin C}}\,\,(dpcm)
\end{array}$
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