Official solution
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.