Mathematics - Determinant Question with Solution | TestHub

$\left[\sin \theta \right]x+\left[-\cos \theta \right]y=0$ and $\left[\cos \theta \right]x+y=0$ for infinite many solution
$\left|\begin{array}{cc}\left[\sin \theta \right]& \left[-\cos \theta \right]\\ \left[\cot \theta \right]& 1\end{array}\right|=0$
i.e. $\left[\sin \theta \right]= \left[-\cos \theta \right] \left[cot\theta \right] ...\left(i\right)$
when $\theta \in \left(\frac{\pi }{2}, \frac{2\pi }{3}\right)$ then $sin\theta \in \left(0, \frac{1}{2}\right) \Rightarrow \left[\sin \theta \right]=0$ and $-\cos \theta \in \left(0, \frac{1}{2}\right) \Rightarrow \left[-\cos \theta \right]=0$ and  

$\cot \theta \in \left(-\frac{1}{\sqrt{3}}, 0\right) \Rightarrow \left[\cot \theta \right]=-1$
When $\theta \in \left(\pi , \frac{7\pi }{6}\right)$ then $sin\theta \in \left(-\frac{1}{2}, 0\right) \Rightarrow \left[\sin \theta \right]=-1$ and $-\cos \theta \in \left(\frac{\sqrt{3}}{2}, 1\right) \Rightarrow \left[-\cos \theta \right]=0$ and 
$\cot \theta \in \left(\sqrt{3}, \infty \right) \Rightarrow \left[\cot \theta \right]\in \left\{2, 3, 4,...\infty \right\}$
When $\theta \in \left(\frac{\pi }{2}, \frac{2\pi }{3}\right)$ then the equation $\left(i\right)$ satisfied therefore infinitely many solution.
When $\theta \in \left(\pi , \frac{7\pi }{6}\right)$ then the equation $\left(i\right)$ not satisfied therefore unique solution.