Solve {l}{y=1/cos(θ)}{tanθ/2=t}

$$\left. \begin{array} { l } { y = \frac { 1 } { \cos ( \theta ) } } \\ { \tan \frac { \theta } { 2 } = t } \end{array} \right.$$

Solve for y, θ, t (complex solution)

$$y=\frac{1}{\cos(\theta )}$$

$$\theta \in \mathrm{C}$$

$$t=\tan(\frac{\theta }{2})$$

$$\nexists n_{2}\in \mathrm{Z}\text{ : }\theta =2\pi n_{2}+\pi \text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}$$

Solve for y, θ, t

$$y=\frac{1}{\cos(\theta )}$$

$$\theta \neq \pi n_{3}+\frac{\pi }{2}$$

$$\forall n_{3}\in \mathrm{Z}$$

$$t\in \mathrm{R}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\frac{\pi \left(2n_{1}+1\right)}{2}\text{ and }\nexists n_{2}\in \mathrm{Z}\text{ : }\theta =\pi \left(2n_{2}+1\right)\text{ and }\exists n_{3}\in \mathrm{Z}\text{ : }\left(\theta >\frac{\pi \left(2n_{3}+1\right)}{2}\text{ and }\theta <\frac{\pi \left(2n_{3}+3\right)}{2}\right)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\theta =2\left(\pi n_{4}+\arcsin(\frac{t}{\sqrt{t^{2}+1}})\right)$$