Question

Solve $\tan 3\theta +\tan \theta =2\tan 2\theta$.

Answer 1

$\tan 3\theta +\tan \theta =2\tan 2\theta$
$\tan 3\theta -\tan 2\theta =\tan 2\theta -\tan \theta$
$\frac{\sin (3\theta -2\theta )}{\cos 3\theta \cos 2\theta }=\frac{\sin (2\theta -\theta )}{\cos 2\theta \cos \theta }$
or $\sin \theta (\cos \theta -\cos 3\theta )=0$
or $\sin \theta \cdot 2\sin \theta \sin 2\theta =0$
$\therefore \sin \theta =0$, $\therefore \theta =n\pi$
$\sin 2\theta =0$, $\therefore 2\theta =n\pi$ or $\theta =\frac{n\pi }{2}$
But $\theta =\frac{n\pi }{2}$ is rejected when n is odd as it makes $\infty =\infty$.
$\therefore \theta =n\pi$ as it will make $0=0$.