If  $\left[x\right]$ denotes the greatest integer $\le x$ , then the system of linear equations  $\left[\sin \theta \right]x+\left[-\cos \theta \right]y=0,\left[\cot \theta \right]x+y=0$

Given system of linear equations is 

$\left[\sin \theta \right]x+\left[-\cos \theta \right]y=0\mathrm{.......}\left(i\right)$

$\text{ And }[\cot \theta ]x+y=0-----\left(ii\right)$

$\text{ Where, }\left[x\right]\text{ denotes the greatest integer }\le x$

$\mathrm{Here} ,\mathrm{\Delta }=\left|\begin{array}{cc}\sin \theta ]& [-\cos \theta ]\\ [\cot \theta ]& 1\end{array}\right|$

$\Rightarrow \Delta =\left[\sin \theta \right]-\left[-\cos \theta \right]\left[\cot \theta \right]$

$\text{ When }\sin \theta \in \left(\frac{\sqrt{3}}{2},1\right)$

$\begin{array}{r}\Rightarrow [\sin \theta ]=0\\ -\cos \theta \in \left(0,\frac{1}{2}\right)\\ \Rightarrow [-\cos \theta ]=0\end{array}$

$and \cot \theta \in \left(-\frac{1}{\sqrt{3}},0\right)$

$\Rightarrow \left[\cot \theta \right]=-1$

$\text{ So, }\mathrm{\Delta }=[\sin \theta ]-[-\cos \theta ][\cot \theta ]$

$-(0\times (-1\left)\right)=0\text{ [ from Eqs. (iii) ,(iv) and }\left(\mathrm{v}\right)\text{ ] }$

$\underset{\_}{\underset{\_}{\text{ Thus, }}}\text{ for }\theta \in \left(\frac{\pi }{2},\frac{2\pi }{3}\right)\text{, the given system have infinitely many solutions. }$

$\text{ When }\theta \in \left(\pi ,\frac{7\pi }{6}\right),\sin \theta \in \left(-\frac{1}{2},0\right)$

$\begin{array}{l}\Rightarrow [\sin \theta ]=-1\\ -\cos \theta \in \left(\frac{\sqrt{3}}{2},1\right)\Rightarrow [\cos \theta ]=0\end{array}$

$\text{ And }\cot \theta \in (\sqrt{3},\mathrm{\infty })\Rightarrow [\cot \theta ]=n,n\in N\text{. }$

$\Delta =-1-\left(0\times n\right)=-1$

$\text{ Thus, for }\theta \in \left(\pi ,\frac{7\pi }{6}\right)\text{, the given system has a unique solution. }$