Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$ - and $z$ -axes, respectively, is half of the angle that this line makes with the positive $x$ -axis. Then the sum of all possible values of the angle $\beta$ is:

Step 1: Define direction cosines and angles

Let the angles that the line makes with the positive $x$ -, $y$ -, and $z$ -axes be $\alpha$, $\beta$, and $\gamma$ respectively. The direction cosines of the line are $\cos \alpha$, $\cos \beta$, and $\cos \gamma$. A fundamental property of direction cosines is that the sum of their squares is equal to 1.

Step 2: Express angles in terms of $\beta$

According to the problem statement, the angle $\beta$ (with the $y$ -axis) and $\gamma$ (with the $z$ -axis) are each half of the angle $\alpha$ (with the $x$ -axis). This means $\beta = \frac{\alpha}{2}$ and $\gamma = \frac{\alpha}{2}$. We can rewrite these relationships as $\alpha = 2\beta$ and $\gamma = \beta$.

Step 3: Substitute into the direction cosine identity

Now, substitute the expressions for $\alpha$ and $\gamma$ in terms of $\beta$ into the direction cosine identity. This gives us an equation solely in terms of $\beta$.

Step 4: Simplify and solve for $\cos \beta$

Using the double angle identity $\cos(2\beta) = 2\cos^2 \beta - 1$, we can simplify the equation. The equation becomes $(2\cos^2 \beta - 1)^2 + 2\cos^2 \beta = 1$. Expanding and rearranging terms, we get $4\cos^4 \beta - 4\cos^2 \beta + 1 + 2\cos^2 \beta = 1$, which simplifies to $4\cos^4 \beta - 2\cos^2 \beta = 0$. Factoring out $2\cos^2 \beta$ yields $2\cos^2 \beta (2\cos^2 \beta - 1) = 0$.

Step 5: Find possible values of $\cos \beta$

From the simplified equation $2\cos^2 \beta (2\cos^2 \beta - 1) = 0$, we have two possibilities: either $2\cos^2 \beta = 0$ or $2\cos^2 \beta - 1 = 0$. If $2\cos^2 \beta = 0$, then $\cos \beta = 0$. If $2\cos^2 \beta - 1 = 0$, then $2\cos^2 \beta = 1$, so $\cos^2 \beta = \frac{1}{2}$, which means $\cos \beta = \pm \frac{1}{\sqrt{2}}$.

Step 6: Determine possible values of $\beta$

For $\cos \beta = 0$, the possible value for $\beta$ in the range $[0, \pi]$ is $\frac{\pi}{2}$. For $\cos \beta = \frac{1}{\sqrt{2}}$, $\beta = \frac{\pi}{4}$. For $\cos \beta = -\frac{1}{\sqrt{2}}$, $\beta = \frac{3\pi}{4}$. These are the possible values for $\beta$ since angles with axes are typically considered in the range $[0, \pi]$. We must also check that $\alpha = 2\beta$ is also in this range. For $\beta = \frac{\pi}{2}$, $\alpha = \pi$. For $\beta = \frac{\pi}{4}$, $\alpha = \frac{\pi}{2}$. For $\beta = \frac{3\pi}{4}$, $\alpha = \frac{3\pi}{2}$, which is outside the range $[0, \pi]$. Therefore, $\beta = \frac{3\pi}{4}$ is not a valid solution.

Step 7: Calculate the sum of possible values of $\beta$

The valid possible values for $\beta$ are $\frac{\pi}{2}$ and $\frac{\pi}{4}$. Summing these values gives the final answer.

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