Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

$\frac{\sin \theta }{\sin \varphi }\cdot \frac{\sin \theta }{\sin \varphi }=\frac{\sin \theta }{\sin \varphi }\cdot \frac{\cos \varphi }{\cos \theta }\Rightarrow \frac{\sin \theta }{\sin \varphi }=\frac{\cos \varphi }{\cos \theta }\Rightarrow \sin 2\theta =\sin 2\varphi$ Since $\tan \theta \ne \tan \varphi ,\theta +\varphi =\frac{\pi }{2}(\mathrm{mod}\pi )$, so $\tan \theta =\cot \varphi$ and $\frac{1}{{\tan }^2\varphi }=3\Rightarrow {\tan }^2\varphi =\frac{1}{3},\tan \varphi =\pm \frac{1}{\sqrt{3}},\tan \theta =\pm \sqrt{3}$