If \( m=\frac{\cos \alpha}{\cos \beta} \) and \( n=\frac{\cos \alpha}{\sin \beta} \), show that ...

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If \( m=\frac{\cos \alpha}{\cos \beta} \) and \( n=\frac{\cos \alpha}{\sin \beta} \), show that \( \left(m^{2}+n^{2}\right) \cos ^{2} \beta=n^{2} \).
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