if $\theta$ $\in$ $\left( {{\pi \over 2},{{2\pi } \over 3}} \right)$ $\Rightarrow \cos \theta \in \left( { - {1 \over 2},0} \right) \Rightarrow - \cos \theta \in \left( {0,{1 \over 2}} \right)$ $\sin \theta \in \left( {{{\sqrt 3 } \over 2},1} \right)$ and $\cot \theta \in \left( { - {1 \over {\sqrt 3 }},0} \right)$ If $\theta \in \left( {\pi ,{{7\pi } \over 6}} \right) \Rightarrow \cos \theta \in \left( { - 1,{{ - \sqrt 3 } \over 2}} \right) \Rightarrow - \cos \theta \in \left( {{{\sqrt 3 } \over 2},1} \right)$ $\sin \theta \in \left( { - {1 \over 2},0} \right)$ and $\cot \theta \in \left( {\sqrt 3 ,\infty } \right)$ Then in $\theta$ $\in$ $\left( {{\pi \over 2},{{2\pi } \over 3}} \right)$ $\Rightarrow$ [sin$\theta$] = 0; [-cos$\theta$] = 0; [cot$\theta$] = -1; Hence 0 = 0 and -x + y = 0 [have infinitely solutions] and in $\theta \in \left( {\pi ,{{7\pi } \over 6}} \right)$ $\Rightarrow$ [sin$\theta$] = -1; [-cos$\theta$] = 0; [cot$\theta$] = 1, 2. 3 ...........; Then -x - 0.y = 0 $\Rightarrow$ x = 0 $[\cot \theta ]x + y = 0$ $\Rightarrow$ 1.x + y = 0 or 2x + y = 0 or ....... each of the line will cut x = 0 at exactly one point. Hence unique solutions.