A line makes an 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^2beta=3sin^2theta,t h e ncos^2theta` equals a. `2/3` b. `1/5` c. `3/5` d. `2/5`

Options

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

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the angles and direction cosines
Given that a line makes an angle \( \theta \) with both the x-axis and z-axis, and an angle \( \beta \) with the y-axis, we can express the direction cosines of the line as:
- \( \cos \theta \) with the x-axis,
- \( \cos \beta \) with the y-axis,
- \( \cos \theta \) with the z-axis.

### Step 2: Write the equation for direction cosines
The relationship between the direction cosines is given by:
\[ \cos^2 \theta + \cos^2 \beta + \cos^2 \theta = 1 \]
This simplifies to:
\[ 2 \cos^2 \theta + \cos^2 \beta = 1 \tag{1} \]

### Step 3: Use the given relationship involving \( \beta \)
We are given that:
\[ \sin^2 \beta = 3 \sin^2 \theta \]
Using the identity \( \sin^2 x = 1 - \cos^2 x \), we can rewrite this as:
\[ 1 - \cos^2 \beta = 3(1 - \cos^2 \theta) \]
Expanding this gives:
\[ 1 - \cos^2 \beta = 3 - 3 \cos^2 \theta \]
Rearranging, we find:
\[ \cos^2 \beta = 3 \cos^2 \theta - 2 \tag{2} \]

### Step 4: Substitute equation (2) into equation (1)
Now we substitute equation (2) into equation (1):
\[ 2 \cos^2 \theta + (3 \cos^2 \theta - 2) = 1 \]
This simplifies to:
\[ 2 \cos^2 \theta + 3 \cos^2 \theta - 2 = 1 \]
Combining like terms gives:
\[ 5 \cos^2 \theta - 2 = 1 \]
Adding 2 to both sides results in:
\[ 5 \cos^2 \theta = 3 \]
Dividing both sides by 5 yields:
\[ \cos^2 \theta = \frac{3}{5} \]

### Step 5: Conclusion
Thus, the value of \( \cos^2 \theta \) is:
\[ \cos^2 \theta = \frac{3}{5} \]
The correct answer is option (c) \( \frac{3}{5} \).

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