A line makes angles $\alpha$, $\beta$, $\gamma$, $\delta$ with the four body diagonals of a cube. What is $\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma + \cos^{2}\delta$?

Explanation

1. 01

   With the edges along the axes, the body diagonals have direction ratios $1, 1, 1$, $-1, 1, 1$, $1, -1, 1$ and $1, 1, -1$, each of length $\sqrt{3}$.

2. 02

   For a line with direction cosines $l, m, n$: $\cos \alpha = \frac{\lvert l + m + n \rvert}{\sqrt{3}}$, and so on for the others.

3. 03

   The four squares add to $\frac{4(l^{2} + m^{2} + n^{2})}{3} = \frac{4}{3}$.