For $k \in \mathbb{R}$, let the solutions of the equation $$\cos \left(\sin ^{-1}\left(x \cot \left(\tan ^{-1}\left(\cos \left(\sin ^{-1} x\right)\right)\right)\right)\right)=k, 0\lt |x|\lt \frac{1}{\sqrt{2}}$$ 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 $\underline{\hspace{2cm}}$.