If for real values of `costheta=x+(1)/(x)`, then

Options

A. `theta` is acute angle
B. `theta` is right angle
C. `theta` is obtuse angle
D. The value of `theta` is not possible

Answer

To solve the problem where we have \( \cos \theta = x + \frac{1}{x} \), we need to determine the values of \( \theta \) based on the real values of \( \cos \theta \).

### Step-by-Step Solution:

1. **Set Up the Equation**:
We start with the equation given:
\[ \cos \theta = x + \frac{1}{x} \]

2. **Rearrange the Equation**:
Multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x \cos \theta = x^2 + 1 \]
Rearranging gives us:
\[ x^2 - x \cos \theta + 1 = 0 \]

3. **Identify the Quadratic Coefficients**:
This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = -\cos \theta \)
- \( c = 1 \)

4. **Use the Discriminant**:
For the quadratic equation to have real roots, the discriminant must be non-negative:
\[ D = b^2 - 4ac \geq 0 \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-\cos \theta)^2 - 4(1)(1) \geq 0 \]
This simplifies to:
\[ \cos^2 \theta - 4 \geq 0 \]

5. **Solve the Inequality**:
Rearranging gives:
\[ \cos^2 \theta \geq 4 \]
Taking the square root of both sides, we find:
\[ |\cos \theta| \geq 2 \]

6. **Analyze the Range of Cosine**:
The range of \( \cos \theta \) for real values of \( \theta \) is:
\[ -1 \leq \cos \theta \leq 1 \]
Since \( |\cos \theta| \geq 2 \) is outside this range, there are no real values of \( \theta \) that satisfy the original equation.

7. **Conclusion**:
Therefore, the value of \( \theta \) is not possible.

### Final Answer:
The value of \( \theta \) is not possible.