If `2costheta=x+1/x`, then `2cos2theta=`?

Answer

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

**Step 1:** Given that \( 2 \cos \theta = x + \frac{1}{x} \).

**Step 2:** We will square both sides of the equation to eliminate the square root when we find \( \cos^2 \theta \):

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

This simplifies to:

\[ 4 \cos^2 \theta = x^2 + 2 + \frac{1}{x^2} \]

**Step 3:** Rearranging gives us:

\[ 4 \cos^2 \theta = x^2 + 2 + \frac{1}{x^2} \]

**Step 4:** We know that \( \cos 2\theta \) can be expressed in terms of \( \cos^2 \theta \) using the double angle formula:

\[ \cos 2\theta = 2 \cos^2 \theta - 1 \]

**Step 5:** Now, we can find \( 2 \cos 2\theta \):

\[ 2 \cos 2\theta = 2(2 \cos^2 \theta - 1) = 4 \cos^2 \theta - 2 \]

**Step 6:** Substitute \( 4 \cos^2 \theta \) from Step 3 into this equation:

\[ 2 \cos 2\theta = (x^2 + 2 + \frac{1}{x^2}) - 2 \]

**Step 7:** Simplifying this gives us:

\[ 2 \cos 2\theta = x^2 + \frac{1}{x^2} \]

Thus, the final result is:

\[ 2 \cos 2\theta = x^2 + \frac{1}{x^2} \]