Question

Solve:$\tan \theta +\tan 2\theta +\tan \theta \tan 2\theta =1$

Answer 1

$\tan \theta +\tan 2\theta +\tan \theta \tan 2\theta =1$

$\Rightarrow \tan \theta +\tan 2\theta =1-\tan \theta \tan 2\theta$

$\Rightarrow \frac{\tan \theta +\tan 2\theta }{1-\tan \theta \tan 2\theta }=1$

$\Rightarrow \tan 3\theta =\tan \frac{\pi }{4}$

$\Rightarrow 3\theta =n\pi +\frac{\pi }{4}$

$\Rightarrow \theta =\frac{n\pi }{3}+\frac{\pi }{12},n\in I$