If `alpha, beta and gamma` are angles which he vector `vec(OP)` (O being the origin) makes with positive direction of the coordinte axes, then which of the following are correct?
1. `cos^(2) alpha + cos^(2) beta = sin^(2) gamma`
2. `sin^(2) alpha + cos^(2)beta= cos^(2)gamma`
3. `sin^(2) alpha + sin^(2) beta + sin^(2) gamma =2`
Select the correct answer the using the codes given below:

Options

A. 1 and 2
B. 2 and 3
C. 1 and 3
D. 1, 2 and 3

Answer

To solve the problem, we need to analyze the three statements given about the angles \( \alpha, \beta, \) and \( \gamma \) that the vector \( \vec{OP} \) makes with the positive direction of the coordinate axes. We will use the properties of direction cosines.

### Step 1: Understand the properties of direction cosines
The direction cosines of a vector are defined as:
- \( \cos \alpha \) = cosine of the angle with the x-axis
- \( \cos \beta \) = cosine of the angle with the y-axis
- \( \cos \gamma \) = cosine of the angle with the z-axis

The fundamental property of direction cosines is:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

### Step 2: Analyze the first statement
The first statement is:
\[ \cos^2 \alpha + \cos^2 \beta = \sin^2 \gamma \]

Using the identity \( \sin^2 \gamma = 1 - \cos^2 \gamma \), we can rewrite the fundamental property:
\[ \cos^2 \alpha + \cos^2 \beta = 1 - \cos^2 \gamma \]
Rearranging gives:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
This implies:
\[ \cos^2 \alpha + \cos^2 \beta = 1 - \cos^2 \gamma \]
Thus, we can conclude:
\[ \cos^2 \alpha + \cos^2 \beta = \sin^2 \gamma \]
So, the first statement is **correct**.

### Step 3: Analyze the second statement
The second statement is:
\[ \sin^2 \alpha + \cos^2 \beta = \cos^2 \gamma \]

Using the identity \( \sin^2 \alpha = 1 - \cos^2 \alpha \), we can rewrite this statement:
\[ 1 - \cos^2 \alpha + \cos^2 \beta = \cos^2 \gamma \]
Rearranging gives:
\[ 1 + \cos^2 \beta - \cos^2 \gamma = \cos^2 \alpha \]
This does not hold true based on the properties of direction cosines. Therefore, the second statement is **incorrect**.

### Step 4: Analyze the third statement
The third statement is:
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \]

Using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we can express this as:
\[ (1 - \cos^2 \alpha) + (1 - \cos^2 \beta) + (1 - \cos^2 \gamma) = 2 \]
This simplifies to:
\[ 3 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) = 2 \]
Since we know \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \), substituting gives:
\[ 3 - 1 = 2 \]
Thus, the third statement is **correct**.

### Conclusion
The correct statements are:
1. \( \cos^2 \alpha + \cos^2 \beta = \sin^2 \gamma \) (Correct)
2. \( \sin^2 \alpha + \cos^2 \beta = \cos^2 \gamma \) (Incorrect)
3. \( \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \) (Correct)

Thus, the correct answer is **Option C: 1 and 3**.