If $\sec\alpha$ and cosec $\alpha$ are the roots of the equation $x^{2}-px+q=0$, then

Roots are sec(a) and cosec(a). Sum of roots = sec(a) + cosec(a) = p. Product of roots = sec(a) * cosec(a) = q. (sec(a) + cosec(a))^2 = p^2 => sec^2(a) + cosec^2(a) + 2*sec(a)*cosec(a) = p^2. Since sec^2(a) + cosec^2(a) = (sin^2(a) + cos^2(a)) / (sin^2(a)cos^2(a)) = 1 / (sin^2(a)cos^2(a)) = sec^2(a)cosec^2(a) = q^2, we have q^2 + 2q = p^2.