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If \( x \) be real, prove that
\[
\frac{x^{2}-2 x \cos \alpha+1}{x^{2}-2 x \cos \beta+1} \text { lies between } \frac{\sin ^{2} \frac{\alpha}{2}}{\sin ^{2} \frac{\beta}{2}} \text { and } \frac{\cos ^{2} \frac{\alpha}{2}}{\cos ^{2} \frac{\beta}{2}} \text {. }
\]
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