Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi }{4}<\beta <0<\alpha <\frac{\pi }{4}$. If  $\sin (\alpha +\beta )=\frac{1}{3}$ and $\cos (\alpha -\beta )=\frac{2}{3}$, then the greatest integer less than or equal to ${(\frac{\sin \alpha }{\cos \beta }+\frac{\cos \beta }{\sin \alpha }+\frac{\cos \alpha }{\sin \beta }+\frac{\sin \beta }{\cos \alpha })}^2$ is 

Given,  $\mathrm{si}(\alpha +\beta )=\frac{1}{3}$
And  $\cos (\alpha -\beta )=\frac{2}{3}$
Let,  $E=\frac{\sin \alpha }{\cos \beta }+\frac{\cos \beta }{\sin \alpha }+\frac{\cos \alpha }{\sin \beta }+\frac{\sin \beta }{\cos \alpha }$
$=\frac{\sin \alpha \sin b+\cos \alpha \cos \beta }{\sin \beta \cos \beta }+\frac{\cos \alpha \cos \beta +\sin \alpha \sin \beta }{\sin \alpha \cos \alpha }$

$=\frac{\cos (\alpha -\beta )}{\sin \beta \cos \beta }+\frac{\cos (\alpha -\beta )}{\sin \alpha \cos \alpha }$

$$\begin{gathered} =\cos (\alpha -\beta )[\frac{2}{2\sin \beta \cos \beta }+\frac{2}{\sin \alpha \cos \alpha }] \\ =\cos (\alpha -\beta )[\frac{2}{2\sin \beta \cos \beta }+\frac{2}{2\sin \alpha \cos \alpha }] \\ =\frac{2}{3}[\frac{2}{\sin 2\beta }+\frac{2}{\sin 2\alpha }] \\ =\frac{4}{3}[\frac{1}{\sin 2\beta }+\frac{1}{\sin 2\alpha }] \\ =\frac{4}{3}\left[\frac{\sin 2\alpha +\sin 2\beta }{\sin 2\alpha \sin 2\beta }\right] \\ =\frac{4\times 2}{3}\left[\frac{2\sin \left(\frac{2\alpha +2\beta }{2}\right)\cos \left(\frac{2\alpha -2\beta }{2}\right)}{2\sin 2\alpha \sin 2\beta }\right] \\ =\frac{16}{3}\left[\frac{\sin (\alpha +\beta )\cos (\alpha -\beta )}{\cos (2\alpha -2\beta )-\cos (2\alpha +2\beta )}\right] \\ =\frac{16}{3}\left[\frac{\frac{1}{3}\times \frac{2}{3}}{(2{\cos }^2(\alpha -\beta )-1)-(1-2{\sin }^2(\alpha +\beta ))}\right] \\ =\frac{32}{27}\left[\frac{1}{2\times \frac{4}{9}-2+2\times \frac{1}{9}}\right] \\ =\frac{32}{27}\left[\frac{9}{8-18+2}\right] \\ =\frac{32}{27}\left[\frac{9}{8-18+2}\right] \\ =\frac{32}{27}\left[\frac{9}{-8}\right] \\ =-\frac{4}{3} \\ \therefore E^2=\frac{16}{9}=1.77 \\ \Rightarrow \left[E^2\right]=1 \end{gathered}$$