Find the number of roots of the equation `tan(x+pi/6)=2tanx ,forx in (0,3pi)dot`

Explanation

To solve the equation \( \tan(x + \frac{\pi}{6}) = 2 \tan x \) for \( x \) in the interval \( (0, 3\pi) \), we will follow these steps:

### Step 1: Use the tangent addition formula
We start with the left side of the equation:
\[ \tan\left(x + \frac{\pi}{6}\right) = \frac{\tan x + \tan\frac{\pi}{6}}{1 - \tan x \tan\frac{\pi}{6}} \]
Since \( \tan\frac{\pi}{6} = \frac{1}{\sqrt{3}} \), we can substitute this value:
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