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  4. The general solution to the equation tan θ+ tan 4 θ+ tan 7 θ= tan θ tan 4 θ tan 7 θ is

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

Trigonometric Functions

Solution:

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 $n \in Z, \text { but } n \neq 6,18,30 \ldots$