Question

The number of solutions of the equation $\tan \theta +\tan 2\theta +\tan 3\theta =\tan \theta \tan 2\theta \tan 3\theta$, if $0<\theta <\pi$ , is

• A. 5
• B. 2
• C. 4
• D. infinite

Answer 1

The correct option is A 5

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

using formula, $\tan (\theta +2\theta +3\theta )=\frac{\tan \theta +\tan 2\theta +\tan 3\theta -\tan \theta .\tan 2\theta .\tan 3\theta }{1-tan\theta .\tan 2\theta -\tan 2\theta .\tan 3\theta -\tan 3\theta .\tan \theta }$

$\Rightarrow \tan 6\theta =0$

Hence solution in the given interval is,
$\theta =\frac{\pi }{6},\frac{2\pi }{6},\frac{3\pi }{6},\frac{4\pi }{6},\frac{5\pi }{6}$

$\therefore$ number of solutions are $5$.