Consider the following relations among angles `alpha,beta and gamma` made by a vector with the coordinate axes.
I. `cos2alpha+cos2beta+cos2gamma=-1`
II. `sin^2alpha+sin^2beta+sin^2gamma=1`
Which of the above statement is/are correct

Options

A. Only I
B. Only II
C. Both I and II
D. Niether I nor II

Answer

To determine the correctness of the given statements regarding the angles \( \alpha, \beta, \) and \( \gamma \) made by a vector with the coordinate axes, we will analyze each statement step by step.

### Step 1: Understanding the Direction Cosines
The direction cosines of a vector with respect to the coordinate axes are defined as:
- \( l = \cos \alpha \) (angle with x-axis)
- \( m = \cos \beta \) (angle with y-axis)
- \( n = \cos \gamma \) (angle with z-axis)

From the properties of direction cosines, we know that:
\[ l^2 + m^2 + n^2 = 1 \]
This can be rewritten as:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \tag{1} \]

### Step 2: Analyzing Statement I
The first statement is:
\[ \cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1 \]

Using the double angle formula for cosine:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
we can express \( \cos 2\alpha, \cos 2\beta, \) and \( \cos 2\gamma \) in terms of \( \cos^2 \alpha, \cos^2 \beta, \) and \( \cos^2 \gamma \):
\[ \cos 2\alpha = 2\cos^2 \alpha - 1 \]
\[ \cos 2\beta = 2\cos^2 \beta - 1 \]
\[ \cos 2\gamma = 2\cos^2 \gamma - 1 \]

Substituting these into the first statement gives:
\[ (2\cos^2 \alpha - 1) + (2\cos^2 \beta - 1) + (2\cos^2 \gamma - 1) = -1 \]
Simplifying this:
\[ 2(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) - 3 = -1 \]
Using equation (1):
\[ 2(1) - 3 = -1 \]
Thus, the first statement is **true**.

### Step 3: Analyzing Statement II
The second statement is:
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1 \]

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express \( \sin^2 \alpha, \sin^2 \beta, \) and \( \sin^2 \gamma \) as:
\[ \sin^2 \alpha = 1 - \cos^2 \alpha \]
\[ \sin^2 \beta = 1 - \cos^2 \beta \]
\[ \sin^2 \gamma = 1 - \cos^2 \gamma \]

Substituting these into the second statement gives:
\[ (1 - \cos^2 \alpha) + (1 - \cos^2 \beta) + (1 - \cos^2 \gamma) = 1 \]
This simplifies to:
\[ 3 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) = 1 \]
Using equation (1):
\[ 3 - 1 = 1 \]
Thus, we have:
\[ 2 = 1 \]
This is **false**.

### Conclusion
Based on the analysis:
- Statement I is **true**.
- Statement II is **false**.

Thus, the correct answer is that **only Statement I is correct**.