Question

Find the general solution of the equation :
​tan x + tan (​​pi/3 + x) + tan (2pi/3 + x) = 3

Answer 1

Dear student
$$\begin{gathered} tanx+tan\left(\frac{\pi }{3}+x\right)+tan\left(\frac{2\pi }{3}+x\right)=3 \\ \Rightarrow tanx+\frac{tanx+tan{\displaystyle \frac{\pi }{3}}}{1-tanxtan{\displaystyle \frac{\pi }{3}}}+\frac{tanx+tan\frac{2\pi }{3}}{1-tanxtan\frac{2\pi }{3}}=3 \\ \Rightarrow tanx+\frac{tanx+\sqrt{3}}{1-\sqrt{3}tanx}+\frac{tanx-\sqrt{3}}{1+\sqrt{3}tanx}=3 \\ \Rightarrow tanx+\frac{8tanx}{1-3ta{n}^{2}x}=3 \\ \Rightarrow \frac{tanx-3ta{n}^{3}x+8tanx}{1-3ta{n}^{2}x}=3 \\ \Rightarrow \frac{3\left(3tanx-ta{n}^{3}x\right)}{1-3ta{n}^{2}x}=3 \\ \Rightarrow 3tan3x=3 \\ \Rightarrow tan3x=1 \\ \Rightarrow tan3x=tan\frac{\pi }{4} \\ \Rightarrow 3x=n\pi +\frac{\pi }{4},n\in Z \\ \Rightarrow x=\frac{n\pi }{3}+\frac{\pi }{12},n\in Z \end{gathered}$$
Regards