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In any triangle \( \mathrm{ABC} \), prove that:
\[
\frac{1+\cos (\mathrm{A}-\mathrm{B}) \cos C}{1+\cos (\mathrm{A}-\mathrm{C}) \cos \mathrm{B}}=\frac{a^{2}+b^{2}}{a^{2}+c^{2}} \text {. }
\]
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