If the direction of the projectile's velocity makes an angle

with the horizontal, then it is true that:
At the point where the projectile hits the ground,
(the velocity vector of the projectile is pointing downwards, so

is in the fourth quadrant).
Now:
 &= 2 \tan \alpha \\ -\tan \beta &= 2 \tan \alpha \\ \tan \beta &= -2 \tan \alpha \\ &= -\frac{gt}{V}\sec \alpha + \tan \alpha\\ &= \frac{\dot y}{\dot x}\end{aligned})