A line makes angles `theta,phi` and `psi` with x,y,z axes respectively. Consider the following.
1. `sin^2theta+sin^2phi=cos^2psi`
2. `cos^2theta+cos^2phi=sin^2psi`
3. `sin^2theta+cos2theta=cos^2psi`
Which of the above is/are correct ?

Options

A. a) 1 only
B. b) 2 only
C. c) 3 only
D. d) 2 and 3

Answer

To determine which of the given statements is correct, we need to analyze each statement based on the properties of direction cosines in 3-D geometry.

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line making angles \( \theta \), \( \phi \), and \( \psi \) with the x-axis, y-axis, and z-axis respectively are given by:
\[ l = \cos \theta, \quad m = \cos \phi, \quad n = \cos \psi \]
According to the properties of direction cosines, we have:
\[ l^2 + m^2 + n^2 = 1 \]
This implies:
\[ \cos^2 \theta + \cos^2 \phi + \cos^2 \psi = 1 \]

2. **Analyzing Statement 1**:
The first statement is:
\[ \sin^2 \theta + \sin^2 \phi = \cos^2 \psi \]
We know that:
\[ \sin^2 \theta = 1 - \cos^2 \theta \quad \text{and} \quad \sin^2 \phi = 1 - \cos^2 \phi \]
Therefore, we can rewrite the left-hand side:
\[ \sin^2 \theta + \sin^2 \phi = (1 - \cos^2 \theta) + (1 - \cos^2 \phi) = 2 - (\cos^2 \theta + \cos^2 \phi) \]
Substituting this into the equation gives:
\[ 2 - (\cos^2 \theta + \cos^2 \phi) = \cos^2 \psi \]
Since \( \cos^2 \theta + \cos^2 \phi + \cos^2 \psi = 1 \), we can rearrange to find:
\[ \cos^2 \theta + \cos^2 \phi = 1 - \cos^2 \psi \]
Thus, the first statement is **not correct**.

3. **Analyzing Statement 2**:
The second statement is:
\[ \cos^2 \theta + \cos^2 \phi = \sin^2 \psi \]
Using the identity \( \sin^2 \psi = 1 - \cos^2 \psi \), we can rewrite the equation:
\[ \cos^2 \theta + \cos^2 \phi = 1 - \cos^2 \psi \]
This is indeed correct as derived from the properties of direction cosines. Therefore, the second statement is **correct**.

4. **Analyzing Statement 3**:
The third statement is:
\[ \sin^2 \theta + \cos^2 \theta = \cos^2 \psi \]
According to the Pythagorean identity, we know:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Therefore, this statement simplifies to:
\[ 1 = \cos^2 \psi \]
This implies \( \cos^2 \psi = 1 \), which means \( \psi = 0 \) or \( \psi = \pi \). This is not generally true for any line making angles with the axes. Thus, the third statement is **not correct**.

### Conclusion:
The only correct statement is the second one.

### Final Answer:
**Correct Statement**: 2 only.