Solve y=tanθ+2/secθ-1

$$y = \frac { \tan \theta + 2 } { \sec \theta - 1 }$$

Solve for y (complex solution)

$$y=-\frac{\left(-2+i\right)e^{i\theta }+\left(-2-i\right)e^{-i\theta }}{2\left(-\cos(\theta )+1\right)}$$

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

Solve for y

$$y=\frac{\sin(\theta )+2\cos(\theta )}{-\cos(\theta )+1}$$

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