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

Let the direction cosines of the straight line be denoted as l, m, and n.

By definition, the direction cosines represent the cosines of the angles a line makes with the positive x, y, and z axes respectively:

$$ l = \cos \alpha $$

$$ m = \cos \beta $$

$$ n = \cos \gamma $$

The problem states that the line makes the same angle $$ \theta $$ with both the x-axis and the z-axis. Therefore:

$$ \alpha = \theta $$

$$ \gamma = \theta $$

Substitute these values to update the direction cosines:

$$ l = \cos \theta $$

$$ m = \cos \beta $$

$$ n = \cos \theta $$

A fundamental identity for direction cosines of any straight line states that the sum of their squares is always equal to 1:

$$ l^2 + m^2 + n^2 = 1 $$

Substitute the expressions for l, m, and n into this property:

$$ \cos^2 \theta + \cos^2 \beta + \cos^2 \theta = 1 $$

Combine the identical terms to simplify the equation:

$$ 2 \cos^2 \theta + \cos^2 \beta = 1 $$

Isolate the term involving the angle $$ \beta $$ on one side:

$$ \cos^2 \beta = 1 - 2 \cos^2 \theta $$

Using the basic trigonometric identity where $$ \sin^2 \beta = 1 - \cos^2 \beta $$, substitute the expression for $$ \cos^2 \beta $$:

$$ \sin^2 \beta = 1 - (1 - 2 \cos^2 \theta) $$

$$ \sin^2 \beta = 2 \cos^2 \theta $$

The question gives a specific condition relating the two angles:

$$ \sin^2 \beta = 3 \sin^2 \theta $$

Equate the two expressions obtained for $$ \sin^2 \beta $$:

$$ 2 \cos^2 \theta = 3 \sin^2 \theta $$

Convert the sine term on the right side into a cosine term using the identity $$ \sin^2 \theta = 1 - \cos^2 \theta $$:

$$ 2 \cos^2 \theta = 3 (1 - \cos^2 \theta) $$

Expand the expression on the right side:

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

Move all the cosine terms to the left side of the equality:

$$ 2 \cos^2 \theta + 3 \cos^2 \theta = 3 $$

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

Divide both sides by 5 to find the final value:

$$ \cos^2 \theta = \frac{3}{5} $$

Final Answer:

The value of $$ \cos^2 \theta $$ is $$ \frac{3}{5} $$.