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  4. If √3 tan 2 θ+√3 tan 3 θ+ tan 2 θ tan 3 θ=1, then the general value of θ is

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Q. If $\sqrt{3} \tan\, 2 \theta+\sqrt{3} \tan\, 3 \theta+\tan \,2 \theta \tan\, 3 \theta=1$, then the general value of $\theta$ is

Trigonometric Functions

Solution:

$\sqrt{3} \tan\, 2 \theta+\sqrt{3} \tan \,3 \theta+\tan\, 2 \theta \tan\, 3 \theta=1$
$\Rightarrow \sqrt{3}(\tan\, 2 \theta+\tan\, 3 \theta)=1-\tan\, 2 \theta \tan \,3 \theta$
$\Rightarrow \frac{\tan\, 2 \theta+\tan \,3 \theta}{1-\tan \,2 \theta \tan\, 3 \theta}=\frac{1}{\sqrt{3}} \Rightarrow \tan \,5 \theta=\tan \frac{\pi}{6}$
$\Rightarrow 5 \theta= n \pi+\frac{\pi}{6} $
$\Rightarrow \theta=\left( n +\frac{1}{6}\right) \frac{\pi}{5}$