Question

Solve:
$\frac{\sin \theta }{\sin 2\theta }=\frac{\cos \theta }{\cos 2\theta }$

Answer 1

$\begin{array}{c}\frac{\sin \theta }{\sin 2\theta }=\frac{\cos \theta }{\cos 2\theta }\\ \Rightarrow \tan \theta =\tan 2\theta \\ \Rightarrow \tan \theta =\frac{2\tan \theta }{1-{\tan }^2\theta }\left[{\tan }^2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }\right]\\ \Rightarrow 1-{\tan }^2\theta =2\\ \Rightarrow {\tan }^2\theta =-1\\ \to {\tan }^2\theta +1=0\\ \Rightarrow {\sec }^2\theta =0\\ \Rightarrow {\cos }^2\theta =0\\ \Rightarrow {\cos }^2\theta ={\cos }^2\left(\frac{\pi }{2}\right)\\ \theta =\left(x\pi \pm \frac{\pi }{2}\right)\end{array}$