Let \( Y = Y(x) \) be the solution of the differential equation \( \sec^{2}x \, dx + \left(e^{2y}\tan^{2}x + \tan x\right) dy = 0, \, 0 < x < \dfrac{\pi}{2}, \, y\left(\dfrac{\pi}{4}\right)=0 \). If \( y\!\left(\dfrac{\pi}{6}\right)=\alpha \), then \( e^{8\alpha} \) is equal to ______.
