Question

Find the general solution of the following equation:

\[\tan 2x \tan x = 1\]

Answer 1

We have:

\[\tan2x \tan x = 1\]

\[\Rightarrow \tan2x = \frac{1}{\tan x}\]

\[ \Rightarrow \tan2x = \cot x\]

\[ \Rightarrow \tan2x = \tan \left( \frac{\pi}{2} - x \right)\]

\[ \Rightarrow 2x = n\pi + \left( \frac{\pi}{2} - x \right), n \in Z\]

\[ \Rightarrow 3x = n\pi + \frac{\pi}{2}, n \in Z\]

\[ \Rightarrow x = \frac{n\pi}{3} + \frac{\pi}{6}, n \in Z\]