If `cos theta =1/2 (x+1/x ) ,` then ` 1/2(x^(2) +1/(x^(2)))` is equal to :

Options

A. `sin2theta`
B. `cos 2theta`
C. `tan 2theta`
D. `sec 2theta`

Answer

To solve the problem, we need to find the value of \( \frac{1}{2} \left( x^2 + \frac{1}{x^2} \right) \) given that \( \cos \theta = \frac{1}{2} \left( x + \frac{1}{x} \right) \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \cos \theta = \frac{1}{2} \left( x + \frac{1}{x} \right) \]

2. **Multiply both sides by 2:**
\[ 2 \cos \theta = x + \frac{1}{x} \]

3. **Square both sides:**
\[ (2 \cos \theta)^2 = \left( x + \frac{1}{x} \right)^2 \]

4. **Expand both sides:**
\[ 4 \cos^2 \theta = x^2 + 2 + \frac{1}{x^2} \]

5. **Rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):**
\[ x^2 + \frac{1}{x^2} = 4 \cos^2 \theta - 2 \]

6. **Now, we need to find \( \frac{1}{2} \left( x^2 + \frac{1}{x^2} \right) \):**
\[ \frac{1}{2} \left( x^2 + \frac{1}{x^2} \right) = \frac{1}{2} \left( 4 \cos^2 \theta - 2 \right) \]

7. **Distributing \( \frac{1}{2} \):**
\[ = 2 \cos^2 \theta - 1 \]

8. **Recognize the double angle identity for cosine:**
\[ 2 \cos^2 \theta - 1 = \cos(2\theta) \]

9. **Thus, we conclude:**
\[ \frac{1}{2} \left( x^2 + \frac{1}{x^2} \right) = \cos(2\theta) \]

### Final Answer:
\[ \frac{1}{2} \left( x^2 + \frac{1}{x^2} \right) = \cos(2\theta) \]

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