Question

Find the general value of $\theta$, if $\sqrt 3 \tan 2\theta + \sqrt 3 \tan 3\theta + \tan 2\theta \tan 3\theta = 1$.(A) $\left[ {n\pi + \dfrac{\pi }{5}} \right]$ (B) $\left( {n + \dfrac{1}{6}} \right)\dfrac{\pi }{5}$ (C) $\left( {2n \pm \dfrac{1}{6}} \right)\dfrac{\pi }{6}$ (D) $\left( {n + \dfrac{1}{3}} \right)\dfrac{\pi }{5}$

Hint: Convert the given equation in equation containing $\tan 5\theta$ using the formula:
$\Rightarrow \tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$
The given equation is $\sqrt 3 \tan 2\theta + \sqrt 3 \tan 3\theta + \tan 2\theta \tan 3\theta = 1$. This can be written as:
$\Rightarrow \sqrt 3 \left( {\tan 2\theta + \tan 3\theta } \right) + \tan 2\theta \tan 3\theta = 1, \\ \Rightarrow \sqrt 3 \left( {\tan 2\theta + \tan 3\theta } \right) = 1 - \tan 2\theta \tan 3\theta , \\ \Rightarrow \sqrt 3 \dfrac{{\left( {\tan 2\theta + \tan 3\theta } \right)}}{{1 - \tan 2\theta \tan 3\theta }} = 1, \\ \Rightarrow \dfrac{{\tan 2\theta + \tan 3\theta }}{{1 - \tan 2\theta \tan 3\theta }} = \dfrac{1}{{\sqrt 3 }} ....(i) \\$
We know that, $\dfrac{1}{{\sqrt 3 }} = \tan \dfrac{\pi }{6}$ and $\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$, if we substitute $A = 2\theta$ and $B = 3\theta$, we have $\dfrac{{\tan 2\theta + \tan 3\theta }}{{1 - \tan 2\theta \tan 3\theta }} = \tan 5\theta$. Putting these values in equation $(i)$, we’ll get:
$\Rightarrow \tan 5\theta = \tan \dfrac{\pi }{6}, \\ \Rightarrow 5\theta = n\pi + \dfrac{\pi }{6}, \\ \Rightarrow \theta = \dfrac{1}{5}\left( {n\pi + \dfrac{\pi }{6}} \right), \\ \Rightarrow \theta = \left( {n + \dfrac{1}{6}} \right)\dfrac{\pi }{5} \\$
Thus, the general value of $\theta$ is $\left( {n + \dfrac{1}{6}} \right)\dfrac{\pi }{5}$. (B) is the correct option.
Note: In the above equation, we get $\tan 5\theta = \tan \dfrac{\pi }{6}$. As we know that $\tan x$ is periodic with period $\pi$, that’s why instead of $5\theta = \dfrac{\pi }{6}$ we have to take $5\theta = n\pi + \dfrac{\pi }{6}$. From this we can say that the value of $5\theta$ will repeat after every integral multiple of $\pi$.