Step-by-Step Solution

1. Let the angles made by the line with the x, y, and z axes be \(\alpha\), \(\beta\), and \(\gamma\) respectively.
2. Given that \(\alpha = \frac{\pi}{4}\) and \(\gamma = \frac{\pi}{4}\). We need to find \(\beta\).
3. We know that \(\cos^2{\alpha} + \cos^2{\beta} + \cos^2{\gamma} = 1\).
4. Substituting the given values, we have \(\cos^2{\frac{\pi}{4}} + \cos^2{\beta} + \cos^2{\frac{\pi}{4}} = 1\).
5. Since \(\cos{\frac{\pi}{4}} = \frac{1}{\sqrt{2}}\), we get \(\left(\frac{1}{\sqrt{2}}\right)^2 + \cos^2{\beta} + \left(\frac{1}{\sqrt{2}}\right)^2 = 1\).
6. This simplifies to \(\frac{1}{2} + \cos^2{\beta} + \frac{1}{2} = 1\).
7. Therefore, \(\cos^2{\beta} = 1 - \frac{1}{2} - \frac{1}{2} = 0\).
8. So, \(\cos{\beta} = 0\).
9. Thus, \(\beta = \frac{\pi}{2}\).