$\cot \theta =\sin 2\theta , \theta \ne n\pi , n\in Z,$ if $\theta$ equals

Solution:

We have

$\begin{array}{l}\cot \theta =\sin 2\theta \\ \Rightarrow \cos \theta =2{\sin }^2\theta \cos \theta \\ \Rightarrow \cos \theta \left(1-2{\sin }^2\theta \right)=0\\ \Rightarrow \cos \theta =0\text{ or, }1-2{\sin }^2\theta =0\\ \Rightarrow \cos \theta =0\text{ or, }{\sin }^2\theta ={\left(\frac{1}{\sqrt{2}}\right)}^2\\ \Rightarrow \theta ={90}^{\circ }\text{ or }\theta ={45}^{\circ }\end{array}$