For k ∈ R, let the solutions of the equation cos ( sin^-1(x cot ( tan^-1( cos ( sin^-1 x)))))=k, 0<|x|<1/(√ 2) be α and β, where the inverse…

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