If cos `theta =1/2 (x+1/x)`, what is the value of cos `2 theta` ?

Answer

To find the value of \( \cos 2\theta \) given that \( \cos \theta = \frac{1}{2} \left( x + \frac{1}{x} \right) \), we can follow these steps:

### Step 1: Use the double angle formula for cosine
The double angle formula for cosine is given by:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]

### Step 2: Substitute the value of \( \cos \theta \)
Substituting the given value of \( \cos \theta \):
\[ \cos 2\theta = 2\left(\frac{1}{2} \left( x + \frac{1}{x} \right)\right)^2 - 1 \]

### Step 3: Simplify the expression
First, calculate \( \left(\frac{1}{2} \left( x + \frac{1}{x} \right)\right)^2 \):
\[ \left(\frac{1}{2} \left( x + \frac{1}{x} \right)\right)^2 = \frac{1}{4} \left( x + \frac{1}{x} \right)^2 \]
Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ = \frac{1}{4} \left( x^2 + 2 + \frac{1}{x^2} \right) = \frac{1}{4} x^2 + \frac{1}{2} + \frac{1}{4} \frac{1}{x^2} \]

### Step 4: Substitute back into the cosine formula
Now substitute back into the cosine formula:
\[ \cos 2\theta = 2\left(\frac{1}{4} x^2 + \frac{1}{2} + \frac{1}{4} \frac{1}{x^2}\right) - 1 \]
Distributing the 2:
\[ = \frac{1}{2} x^2 + 1 + \frac{1}{2} \frac{1}{x^2} - 1 \]

### Step 5: Simplify the expression
Now simplify:
\[ \cos 2\theta = \frac{1}{2} x^2 + \frac{1}{2} \frac{1}{x^2} \]
Thus, we have:
\[ \cos 2\theta = \frac{1}{2} \left( x^2 + \frac{1}{x^2} \right) \]

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

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