Solve u^2-(alpha+beta)sec(p)=0

$$u ^ { 2 } - ( \alpha + \beta ) \sec ( p ) = 0$$

Solve for u (complex solution)

$$u=-\frac{\sqrt{2}i\sqrt{2\left(-\alpha -\beta \right)}\left(\cos(p)\right)^{-\frac{1}{2}}}{2}$$

$$u=\frac{\sqrt{2}i\sqrt{2\left(-\alpha -\beta \right)}\left(\cos(p)\right)^{-\frac{1}{2}}}{2}\text{, }\nexists n_{1}\in \mathrm{Z}\text{ : }p=\pi n_{1}+\frac{\pi }{2}$$

Solve for u

$$u=\sqrt{\frac{\alpha +\beta }{\cos(p)}}$$

$$u=-\sqrt{\frac{\alpha +\beta }{\cos(p)}}\text{, }\left(\alpha =-\beta \text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }p=\pi n_{1}+\frac{\pi }{2}\right)\text{ or }\left(\alpha \leq -\beta \text{ and }\exists n_{3}\in \mathrm{Z}\text{ : }\left(p>2\pi n_{3}+\frac{\pi }{2}\text{ and }p<2\pi n_{3}+\frac{3\pi }{2}\right)\right)\text{ or }\left(\alpha \geq -\beta \text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\left(p>2\pi n_{2}+\frac{3\pi }{2}\text{ and }p<2\pi n_{2}+\frac{5\pi }{2}\right)\right)$$