For \( k \in \mathbb{R} \), let the solutions of the equation \[ \cos \left(\sin ^{-1}\left(x \c...

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For \( k \in \mathbb{R} \), let the solutions of the equation
\[
\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right)\right)\right)=k, 0|x|\frac{1}{\sqrt{2}}
\]
be \( \alpha \) and \( \beta \), where the inverse trigonometric functions take only principal values. If the solutions of the equation \( x^{2}- \) \( b x-5=0 \) are \( \frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}} \) and \( \frac{\alpha}{\beta} \), then \( \frac{b}{k^{2}} \) is equal to
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