Given:

Each of the angles $\beta$ and $\gamma$ is half of the angle that the line makes with the positive $x$-axis, i.e., $\beta = \gamma = \frac{\alpha}{2}$.

The equation for the direction cosines of angles a line makes with the coordinate axes is given by:

$ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 $

Since $\beta = \gamma$, we substitute to get:

$ \cos^2 \alpha + 2 \cos^2 \beta = 1 $

Substitute $\cos \beta = \cos \gamma$:

$ \cos \alpha = 2 \cos^2 \beta - 1 $

Replacing back into the equation:

$ (2 \cos^2 \beta - 1)^2 + 2 \cos^2 \beta = 1 $

Simplify:

$ (2 \cos^2 \beta - 1)(2 \cos^2 \beta + 1) = 0 $

Solving for $\cos^2 \beta$, we have:

$ 2 \cos^2 \beta - 1 = 0 \quad \text{or} \quad 2 \cos^2 \beta + 1 = 0 $

The latter gives no real solutions, thus:

$ \cos^2 \beta = \frac{1}{2} $

Therefore, $\beta = \frac{\pi}{4}$ or $\beta = \frac{\pi}{2}$.

Thus, the sum of all possible values of $\beta$ is:

$ \frac{\pi}{4} + \frac{\pi}{2} = \frac{3\pi}{4} $