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 $\_\_\_\_$ .
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 $\_\_\_\_$ .