Question

The general solution to the equation $\tan \theta +\tan 4\theta +\tan 7\theta =\tan \theta \tan 4\theta \tan 7\theta$ is

• A. $\theta =n\pi /12$, where $n\in Z$
• B. $\theta =n\pi /9$, where $n\in Z$
• C. $\theta =n\pi +\pi /12$, where $n\in Z$
• D. None of these

Answer 1

Answer: (D)

From the given equation, we have
$\frac{\tan \theta +\tan 4\theta }{1-\tan \theta \tan 4\theta }=-\tan 7\theta$
$\Rightarrow \tan (\theta +4\theta )=-\tan 7\theta$
$\Rightarrow \tan 5\theta =\tan (-7\theta )$
$\Rightarrow 5\theta =n\pi -7\theta$
$\Rightarrow \theta =n\pi 12,$ where $n\in Z,\text{ but }n\ne 6,18,30\dots$