Solve y=tan(x+pi/6)

$$y= \tan ( x+ \frac{ \pi }{ 6 } )$$

Solve for x

$$x=\frac{12\pi n_{1}+6\arcsin(\frac{y}{\sqrt{y^{2}+1}})+5\pi }{6}\text{, }n_{1}\in \mathrm{Z}\text{, }\exists n_{3}\in \mathrm{Z}\text{ : }\left(n_{1}>\frac{2n_{3}-\frac{2\arcsin(\frac{y}{\sqrt{y^{2}+1}})}{\pi }-1}{4}\text{ and }n_{1}<\frac{2n_{3}-\frac{2\arcsin(\frac{y}{\sqrt{y^{2}+1}})}{\pi }+1}{4}\right)$$

$$x=\frac{12\pi n_{2}+6\arcsin(\frac{y}{\sqrt{y^{2}+1}})-\pi }{6}\text{, }n_{2}\in \mathrm{Z}\text{, }\exists n_{3}\in \mathrm{Z}\text{ : }\left(n_{3}>\frac{4n_{2}+\frac{2\arcsin(\frac{y}{\sqrt{y^{2}+1}})}{\pi }-3}{2}\text{ and }n_{3}<\frac{4n_{2}+\frac{2\arcsin(\frac{y}{\sqrt{y^{2}+1}})}{\pi }-1}{2}\right)$$

Solve for y

$$y=\frac{\sqrt{3}\sin(x)+\cos(x)}{\sqrt{3}\cos(x)-\sin(x)}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{3}$$