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  4. If tan θ+ tan 4 θ+ tan 7 θ= tan θ tan 4 θ tan 7 θ, then θ=

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Q. If $\tan \theta+\tan 4 \theta+\tan 7 \theta=\tan \theta \tan 4 \theta \tan 7 \theta$, then $\theta=$

Trigonometric Functions

Solution:

$\tan (7 \theta+4 \theta+\theta)=\frac{\tan 7 \theta+\tan 4 \theta+\tan \theta-\tan 7 \theta \tan 4 \theta \tan \theta}{1-\tan 7 \theta \tan 4 \theta-\tan 4 \theta \tan \theta-\tan \theta \tan 7 \theta} $
$\Rightarrow \tan 12 \theta=0 $
$\therefore 12 \theta= n \pi, \theta=\frac{ n \pi}{12}$
$\text { clearly } \frac{ n \pi}{12} \neq(2 k +1) \frac{\pi}{2} $
$\Rightarrow n \neq 6(2 k +1)$
n. $k \in I$