Key Concepts and Formulas

• Direction Cosines Identity: For any line making angles $\alpha, \beta, \gamma$ with the positive $x, y, z$-axes respectively, the sum of the squares of its direction cosines is always 1: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ The angles $\alpha, \beta, \gamma$ are conventionally taken in the range $[0, \pi]$.
• Trigonometric Double Angle Identity: The identity relating $\cos \theta$ and $\cos (2\theta)$ is crucial: $2\cos^2 \theta = 1 + \cos (2\theta)$ This can be rewritten as $\cos^2 \theta = \frac{1 + \cos (2\theta)}{2}$.

Step-by-Step Solution

• Step 1: Set up the relationships between the angles. The problem states that each of the angles $\beta$ and $\gamma$ is half of the angle $\alpha$.

  • What we are doing: Translating the given problem statement into mathematical equations.
  • Why we are doing it: To express all angles in terms of a single variable, which simplifies the direction cosines identity.
  • Math: Let the angle with the positive $x$-axis be $\alpha$. Given, the angle with the positive $y$-axis is $\beta$, and the angle with the positive $z$-axis is $\gamma$. According to the problem: $\beta = \frac{\alpha}{2}$ $\gamma = \frac{\alpha}{2}$ From these relations, it is clear that $\beta = \gamma$.

• Step 2: Substitute these relationships into the direction cosines identity.

  • What we are doing: Using the fundamental identity of direction cosines and substituting the expressions for $\beta$ and $\gamma$ from Step 1.
  • Why we are doing it: To form a single trigonometric equation in terms of $\alpha$ that we can solve.
  • Math: The direction cosines identity is $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Substituting $\beta = \frac{\alpha}{2}$ and $\gamma = \frac{\alpha}{2}$: $\cos^2 \alpha + \cos^2 \left(\frac{\alpha}{2}\right) + \cos^2 \left(\frac{\alpha}{2}\right) = 1$ $\cos^2 \alpha + 2\cos^2 \left(\frac{\alpha}{2}\right) = 1$

• Step 3: Solve the trigonometric equation for $\cos \alpha$.

  • What we are doing: Applying a trigonometric identity to simplify the equation and solve for $\cos \alpha$.
  • Why we are doing it: The equation contains both $\cos \alpha$ and $\cos(\alpha/2)$, so we use the double angle identity $2\cos^2 \theta = 1 + \cos(2\theta)$ (with $\theta = \alpha/2$) to express everything in terms of $\cos \alpha$.
  • Math: Using the identity $2\cos^2 \left(\frac{\alpha}{2}\right) = 1 + \cos \alpha$, substitute this into the equation from Step 2: $\cos^2 \alpha + (1 + \cos \alpha) = 1$ Rearrange the terms: $\cos^2 \alpha + \cos \alpha = 0$ Factor out $\cos \alpha$: $\cos \alpha (\cos \alpha + 1) = 0$ This equation yields two possible cases for $\cos \alpha$:
    1. $\cos \alpha = 0$
    2. $\cos \alpha + 1 = 0 \implies \cos \alpha = -1$

• Step 4: Determine possible values of $\alpha$ and corresponding $\beta$.

  • What we are doing: Finding the angles $\alpha$ that satisfy the conditions, and then the corresponding values of $\beta$.

  • Why we are doing it: We need to find all valid $\beta$ values to sum them up. We consider the standard range for angles a line makes with axes, which is $\alpha \in [0, \pi]$.

  • Math and Reasoning:

    • Case 1: $\cos \alpha = 0$ For $\alpha \in [0, \pi]$, $\cos \alpha = 0$ implies $\alpha = \frac{\pi}{2}$. Then, $\beta = \frac{\alpha}{2} = \frac{(\pi/2)}{2} = \frac{\pi}{4}$. Let's check if this is a valid set of angles: $\cos^2(\pi/2) + \cos^2(\pi/4) + \cos^2(\pi/4) = 0^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = 0 + \frac{1}{2} + \frac{1}{2} = 1$. This is valid.

    • Case 2: $\cos \alpha = -1$ For $\alpha \in [0, \pi]$, $\cos \alpha = -1$ implies $\alpha = \pi$. Then, $\beta = \frac{\alpha}{2} = \frac{\pi}{2}$. Let's check if this is a valid set of angles: $\cos^2(\pi) + \cos^2(\pi/2) + \cos^2(\pi/2) = (-1)^2 + 0^2 + 0^2 = 1 + 0 + 0 = 1$. This is valid.

    So, we have two possible values for $\beta$: $\frac{\pi}{4}$ and $\frac{\pi}{2}$.

• Step 5: Consider implicit conditions and sum the possible values of $\beta$.

  • What we are doing: Analyzing the possible values for $\beta$ in light of potential implicit conditions in the problem statement.

  • Why we are doing it: Sometimes, in geometry problems, phrasing like "the angle a line makes with an axis" can implicitly suggest that the line is not perpendicular to that axis, implying a non-zero projection. If the line is perpendicular to the x-axis (i.e., $\alpha = \pi/2$), its projection onto the x-axis is zero ($\cos \alpha = 0$). If this case is implicitly excluded, then $\alpha = \pi/2$ (and thus $\beta = \pi/4$) would not be a valid solution. Assuming this implicit condition (that $\cos \alpha \ne 0$), we exclude the first case.

  • Math: Excluding the case where $\cos \alpha = 0$ (i.e., $\alpha = \pi/2$) leaves us with only one valid scenario: $\alpha = \pi$, which leads to $\beta = \frac{\pi}{2}$. Therefore, the only possible value for $\beta$ under this interpretation is $\frac{\pi}{2}$.

    The sum of all possible values of $\beta$ is simply $\frac{\pi}{2}$.

Common Mistakes & Tips

• Forgetting the Range of Angles: Remember that angles a line makes with the axes ($\alpha, \beta, \gamma$) are conventionally in the range $[0, \pi]$. This helps in uniquely determining $\alpha$ from $\cos \alpha$.
• Trigonometric Identity Errors: Be careful with the double angle identity. A common mistake is to write $2\cos^2(\theta) = \cos(2\theta) - 1$. The correct form is $2\cos^2(\theta) = 1 + \cos(2\theta)$.
• Implicit Assumptions: In competitive exams, sometimes subtle phrasing implies additional conditions. If your initial mathematical derivation gives multiple answers, and only one leads to an option, re-evaluate if any of your derived cases could be implicitly excluded (e.g., perpendicularity leading to a zero direction cosine).

Summary The problem leverages the fundamental identity of direction cosines, $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. By expressing $\beta$ and $\gamma$ in terms of $\alpha$ as $\alpha/2$, the identity transforms into a trigonometric equation in $\cos \alpha$. Solving this equation, $\cos \alpha (\cos \alpha + 1) = 0$, yields $\alpha = \pi/2$ or $\alpha = \pi$. These give possible $\beta$ values of $\pi/4$ and $\pi/2$. However, if we implicitly assume that the line is not perpendicular to the x-axis (i.e., $\cos \alpha \ne 0$), then $\alpha = \pi/2$ is excluded. This leaves $\alpha = \pi$ as the only valid solution, leading to $\beta = \pi/2$. Thus, the sum of all possible values for $\beta$ is $\pi/2$.

Final Answer The final answer is $\boxed{\frac{\pi}{2}}$ which corresponds to option (A).