Question

General solution of $\tan \theta +\tan 4\theta +\tan 7\theta =\tan \theta \tan 4\theta \tan 7\theta$ is

• A. $\theta =n\frac{\pi }{12},$ where $nϵz$
• B. $\theta =n\frac{\pi }{9},$ where $nϵz$
• C. $\theta =n\pi +\frac{\pi }{12}$,where $nϵz$
• D. None of these

Answer 1

The correct option is A $\theta =n\frac{\pi }{12},$ where $nϵz$

Take $\tan 7\theta$ to LHS.

$\tan \theta +\tan 4\theta =\tan 7\theta (\tan \theta \tan 4\theta -1)$

$\therefore \frac{\tan \theta +\tan 4\theta }{\tan \theta \tan 4\theta -1}=\tan 7\theta$

$\therefore -\left(\tan 5\theta \right)=\tan 7\theta$

$\therefore \left(\tan 5\theta \right)=\tan (n\pi -7\theta )$

$\therefore 5\theta =n\pi -7\theta$

$\therefore 12\theta =n\pi$

$\therefore \theta =\frac{n\pi }{12}$