A line makes the same angle $\theta$ with each of the $x$ and $z$-axes. If the angle $\beta$, which it makes with the $y$-axis is such that $\sin ^2 \beta=3 \sin ^2 \theta$, then $\cos ^2 \theta$ equals

Let the direction cosines of the line be $l, m, n$. Since the line makes the same angle $\theta$ with the $x$ and $z$-axes, we have $l = \cos \theta$ and $n = \cos \theta$.

Let $\beta$ be the angle the line makes with the $y$-axis, so $m = \cos \beta$.

The sum of the squares of the direction cosines is $l^2 + m^2 + n^2 = 1$.

Substituting the values, we get $\cos^2 \theta + \cos^2 \beta + \cos^2 \theta = 1$, which simplifies to $2 \cos^2 \theta + \cos^2 \beta = 1$.

We know that $\sin^2 \beta = 1 - \cos^2 \beta$ and $\sin^2 \theta = 1 - \cos^2 \theta$.

Given the condition $\sin^2 \beta = 3 \sin^2 \theta$, we substitute the identities: $1 - \cos^2 \beta = 3(1 - \cos^2 \theta)$.

From the relation $2 \cos^2 \theta + \cos^2 \beta = 1$, we have $\cos^2 \beta = 1 - 2 \cos^2 \theta$.

Substituting this into the condition: $1 - (1 - 2 \cos^2 \theta) = 3 - 3 \cos^2 \theta$.

$2 \cos^2 \theta = 3 - 3 \cos^2 \theta$.

$5 \cos^2 \theta = 3$.

Therefore, $\cos^2 \theta = \dfrac{3}{5}$.

Answer: $\dfrac{3}{5}$