Given \( x+\frac{1}{x}=3 \cos A \) and \( x^{2}+\frac{1}{x^{2}}=2 \) Determine the value of \( \cos 2 \mathrm{~A} \) without using a calculator 2

We start with the given equations: \[ x+\frac{1}{x}=3\cos A \quad \text{and} \quad x^2+\frac{1}{x^2}=2. \] **Step 1. Express \( x^2+\frac{1}{x^2} \) in terms of \(x+\frac{1}{x}\):** We know that \[ \left(x+\frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}. \] Thus, isolating \( x^2+\frac{1}{x^2} \) gives \[ x^2+\frac{1}{x^2} = \left(x+\frac{1}{x}\right)^2 - 2. \] **Step 2. Substitute the given expression for \( x+\frac{1}{x} \):** Since \( x+\frac{1}{x}=3\cos A \), substitute this into the previous equation: \[ x^2+\frac{1}{x^2} = (3\cos A)^2 - 2 = 9\cos^2 A - 2. \] **Step 3. Set equal to the given value of \( x^2+\frac{1}{x^2} \):** We are given that \[ x^2+\frac{1}{x^2} = 2. \] Thus, \[ 9\cos^2 A - 2 = 2. \] **Step 4. Solve for \( \cos^2 A \):** Add \(2\) to both sides: \[ 9\cos^2 A = 4. \] Then, \[ \cos^2 A = \frac{4}{9}. \] **Step 5. Use the double angle formula for cosine:** The double angle formula is given by \[ \cos 2A = 2\cos^2 A - 1. \] Substitute \(\cos^2 A =\frac{4}{9}\) into the formula: \[ \cos 2A = 2\left(\frac{4}{9}\right) - 1 = \frac{8}{9} - 1. \] **Step 6. Simplify to find \( \cos 2A \):** \[ \cos 2A = \frac{8}{9} - \frac{9}{9} = -\frac{1}{9}. \] The value of \( \cos 2A \) is therefore \[ -\frac{1}{9}. \]