A line makes the same angle $\theta$ with the $x$- and $z$-axes and an angle $\beta$ with the $y$-axis, where $\sin^{2}\beta = 3\sin^{2}\theta$. What is $\cos^{2}\theta$?

Explanation

1. 01

   $\cos^{2}\beta = 1 - \sin^{2}\beta = 1 - 3(1 - \cos^{2}\theta) = 3\cos^{2}\theta - 2$.

2. 02

   The squared direction cosines add to 1: $2\cos^{2}\theta + \cos^{2}\beta = 5\cos^{2}\theta - 2 = 1$, so $\cos^{2}\theta = \frac{3}{5}$.