If \( \tan \alpha=n \tan \beta \) and \( \sin \alpha=m \sin \beta \...

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If \( \tan \alpha=n \tan \beta \) and \( \sin \alpha=m \sin \beta \), prove that:
\[
\cos ^{2} \alpha=\frac{m^{2}-1}{n^{2}-1} \text {. }
\]
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