$\left[ {\cot \theta } \right]x + y = 0\,\,\,\,\,\,\,......\left( 2 \right)$

Case $I$

Whene $\theta  \in \left( {\frac{\pi }{2},\frac{{2\pi }}{3}} \right)$

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

$\cos \theta  \in \left( { - \frac{1}{2},0} \right) - \cos \theta  \in \left( {0,\frac{1}{2}} \right)$

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

$\left[ {\sin \theta } \right] = 0\,\,\,\,\,\left[ { - \cos \theta } \right] = 0\,\,\,\,\,\left[ {\cot \theta } \right] =  - 1$

Equation $(1)$ and $(2)$ will

$\left. \begin{array}{l}
0x + 0y = 0\\
 - x + y = 0
\end{array} \right]$ ystem will have infinitely many solution 

Case $II$

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

$\cos \theta  \in \left( { - 1,\frac{{\sqrt 3 }}{2}} \right)$

$\cot \theta  \in \left( {\sqrt 3 ,\infty } \right)$

$\left[ {\sin \theta } \right] =  - 1,\left[ {\cos \theta } \right] =  - 1$

$\left[ {\cot \theta } \right] = \left\{ {1,2,3,......} \right\}$

$-x-y=0$

$Ix+y=0$              I={1,2,.....}

It will have unique solution in all cases $x=0,y=0$