If `2costheta=x+(1)/(x)`, then find the value of `2cos^(2)theta`.

Options

A. a) `(1)/(2)(x^(2)+(1)/(x^(2)))+1`
B. b) `(x^(2)+(1)/(x^(2)))-1`
C. c) `(3)/(x^(3))`
D. d) 1

Answer

To solve the problem, we start with the given equation:

1. **Given Equation:**
\[ 2 \cos \theta = x + \frac{1}{x} \]

2. **Square Both Sides:**
We will square both sides of the equation to eliminate the square root later:
\[ (2 \cos \theta)^2 = \left(x + \frac{1}{x}\right)^2 \]
This gives us:
\[ 4 \cos^2 \theta = x^2 + 2 + \frac{1}{x^2} \]

3. **Rearranging the Equation:**
We can rearrange the equation to isolate \(4 \cos^2 \theta\):
\[ 4 \cos^2 \theta = x^2 + 2 + \frac{1}{x^2} \]

4. **Divide by 2:**
Now, we will divide the entire equation by 2 to find \(2 \cos^2 \theta\):
\[ 2 \cos^2 \theta = \frac{x^2 + 2 + \frac{1}{x^2}}{2} \]

5. **Final Expression for \(2 \cos^2 \theta\):**
We can simplify this expression:
\[ 2 \cos^2 \theta = \frac{1}{2} \left(x^2 + \frac{1}{x^2}\right) + 1 \]

Thus, the value of \(2 \cos^2 \theta\) is:
\[ 2 \cos^2 \theta = \frac{1}{2} \left(x^2 + \frac{1}{x^2}\right) + 1 \]