Solve tanx=frac{sqrt{t^2-9}}{3}

$$\tan x = \frac { \sqrt { t ^ { 2 } - 9 } } { 3 }$$

Solve for t (complex solution)

$$t=-6\times \left(2\left(\cos(2x)+1\right)\right)^{-\frac{1}{2}}$$

$$t=6\times \left(2\left(\cos(2x)+1\right)\right)^{-\frac{1}{2}}\text{, }\left(\nexists n_{3}\in \mathrm{Z}\text{ : }x=\frac{\pi n_{3}}{2}\text{ and }arg(\tan(x))<\pi \right)\text{ or }\left(\exists n_{2}\in \mathrm{Z}\text{ : }x=\pi n_{2}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }x=\frac{\pi \left(2n_{1}+1\right)}{2}\right)$$

Solve for t

$$t=\frac{3}{\cos(x)}$$

$$t=-\frac{3}{\cos(x)}\text{, }\exists n_{1}\in \mathrm{Z}\text{ : }\left(x\geq \pi n_{1}\text{ and }x<\pi n_{1}+\frac{\pi }{2}\right)$$

Solve for x

$$x=\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{14}\text{, }n_{14}\in \mathrm{Z}\text{, }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=1\text{ and }n_{2}=-2+2n_{14}\right)\text{ or }\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=1\text{ and }n_{2}=2n_{14}\right)\text{ or }\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=1\text{ and }not(n_{2}>2n_{14})\text{ and }not(n_{2}<-2+2n_{14})\right)\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{14}=\frac{1}{2}\pi +\pi n_{1}$$

$$x=\pi +2\pi n_{13}+arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})\text{, }n_{13}\in \mathrm{Z}\text{, }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=0\text{ and }n_{2}=-1+2n_{13}\right)\text{ or }\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=0\text{ and }n_{2}=1+2n_{13}\right)\text{ or }\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=0\text{ and }not(n_{2}>1+2n_{13})\text{ and }not(n_{2}<-1+2n_{13})\right)\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2\pi n_{13}+arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})=\frac{1}{2}\pi +\pi n_{1}$$

$$x=arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{34}\text{, }n_{34}\in \mathrm{Z}\text{, }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=1\text{ and }n_{2}=\left(-1\right)\left(2+\left(-2\right)n_{34}\right)\right)\text{ or }\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=1\text{ and }n_{2}=2n_{34}\right)\text{ or }\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=1\text{ and }not(n_{2}>2n_{34})\text{ and }not(n_{2}<\left(-1\right)\left(2+\left(-2\right)n_{34}\right))\right)\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{34}=\frac{1}{2}\pi +\pi n_{1}$$

$$x=\pi +2n_{33}\pi +\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})\text{, }n_{33}\in \mathrm{Z}\text{, }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\left(\left(not(|t|<3)\text{ and }\nexists n_{85}\in \mathrm{Z}\text{ : }n_{2}=1+\left(-2\right)n_{85}\text{ and }not(n_{2}>1+2n_{33})\right)\text{ or }\left(not(n_{2}>2n_{33}-1)\text{ and }not(|t|<3)\right)\right)\text{ and }\left(\left(not(|t|<3)\text{ and }n_{2}\text{bmod}2=0\text{ and }not(n_{2}<-1+2n_{33})\right)\text{ or }\left(not(n_{2}<2n_{33}+1)\text{ and }not(|t|<3)\right)\right)\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2n_{33}\pi +\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})=\frac{1}{2}\pi +\pi n_{1}$$

$$x=\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{6}\text{, }n_{6}\in \mathrm{Z}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{6}=\frac{1}{2}\pi +\pi n_{1}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{6}=\frac{1}{2}\pi +\pi n_{1}$$

$$x=\pi +2\pi n_{7}+arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})\text{, }n_{7}\in \mathrm{Z}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2\pi n_{7}+arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})=\frac{1}{2}\pi +\pi n_{1}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2\pi n_{7}+arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})=\frac{1}{2}\pi +\pi n_{1}$$

$$x=arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{26}\text{, }n_{26}\in \mathrm{Z}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{26}=\frac{1}{2}\pi +\pi n_{1}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})+2\pi n_{26}=\frac{1}{2}\pi +\pi n_{1}$$

$$x=\pi +2n_{27}\pi +\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})\text{, }n_{27}\in \mathrm{Z}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2n_{27}\pi +\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})=\frac{1}{2}\pi +\pi n_{1}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2n_{27}\pi +\left(-1\right)arcSin(\left(t^{2}-9\right)^{\frac{1}{2}}t^{-1})=\frac{1}{2}\pi +\pi n_{1}$$