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

Options

A. `2/5`
B. `1/5`
C. `3/5`
D. `2/3`

Explanation

To solve the problem step by step, we start with the information given in the question:

1. A line makes the same angle \( \theta \) with the \( x \)-axis and \( z \)-axis.
2. The angle \( \beta \) with the \( y \)-axis is such that \( \sin^2 \beta = 3 \sin^2 \theta \).

### Step 1: Understand the relationship between angles and cosines
We know that for any line making angles \( \alpha \), \( \beta \), and \( \gamma \) with the \( x \)-, \( y \)-, and \( z \)-axes respectively, the following relationship holds:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
In our case, since the line makes the same angle \( \theta \) with the \( x \)- and \( z \)-axes, we have:
\[ \alpha = \theta, \quad \gamma = \theta, \quad \beta = \beta \]
Thus, we can rewrite the equation as:
\[ \cos^2 \theta + \cos^2 \beta + \cos^2 \theta = 1 \]
This simplifies to:
\[ 2 \cos^2 \theta + \cos^2 \beta = 1 \]

### Step 2: Express \( \cos^2 \beta \) in terms of \( \cos^2 \theta \)
From the equation we derived:
\[ \cos^2 \beta = 1 - 2 \cos^2 \theta \]

### Step 3: Use the given relationship for \( \sin^2 \beta \)
We are given that:
\[ \sin^2 \beta = 3 \sin^2 \theta \]
Using the identity \( \sin^2 \beta = 1 - \cos^2 \beta \), we can substitute for \( \sin^2 \beta \):
\[ 1 - \cos^2 \beta = 3 \sin^2 \theta \]
Substituting \( \cos^2 \beta \) from Step 2 into this equation gives:
\[ 1 - (1 - 2 \cos^2 \theta) = 3 \sin^2 \theta \]
This simplifies to:
\[ 2 \cos^2 \theta = 3 \sin^2 \theta \]

### Step 4: Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \)
We can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Substituting this into the equation from Step 3 gives:
\[ 2 \cos^2 \theta = 3(1 - \cos^2 \theta) \]
Expanding and rearranging:
\[ 2 \cos^2 \theta = 3 - 3 \cos^2 \theta \]
\[ 2 \cos^2 \theta + 3 \cos^2 \theta = 3 \]
\[ 5 \cos^2 \theta = 3 \]
Thus, we find:
\[ \cos^2 \theta = \frac{3}{5} \]

### Final Answer
\[ \cos^2 \theta = \frac{3}{5} \]