If ` cos 2 theta =(sqrt(2)+1)( cos theta -(1)/(sqrt(2)))`, then the value of ` theta ` is

Options

A. `2 n pi +(pi)/(4)`
B. `2n pi pm (pi)/4`
C. `2npi -(pi)/4`
D. None of these

Answer

To solve the equation \( \cos 2\theta = (\sqrt{2}+1)\left(\cos \theta - \frac{1}{\sqrt{2}}\right) \), we will follow these steps:

### Step 1: Substitute the double angle formula
We know that \( \cos 2\theta = 2\cos^2 \theta - 1 \). Substituting this into the equation gives us:
\[ 2\cos^2 \theta - 1 = (\sqrt{2}+1)\left(\cos \theta - \frac{1}{\sqrt{2}}\right) \]

### Step 2: Simplify the right-hand side
Distributing the right-hand side:
\[ 2\cos^2 \theta - 1 = (\sqrt{2}+1)\cos \theta - \frac{\sqrt{2}+1}{\sqrt{2}} \]
The term \( \frac{\sqrt{2}+1}{\sqrt{2}} \) simplifies to \( 1 + \frac{1}{\sqrt{2}} \).

### Step 3: Rearranging the equation
Now, we rewrite the equation:
\[ 2\cos^2 \theta - 1 = (\sqrt{2}+1)\cos \theta - 1 - \frac{1}{\sqrt{2}} \]
Adding \( 1 + \frac{1}{\sqrt{2}} \) to both sides gives:
\[ 2\cos^2 \theta = (\sqrt{2}+1)\cos \theta + \frac{1}{\sqrt{2}} \]

### Step 4: Move all terms to one side
Rearranging gives us:
\[ 2\cos^2 \theta - (\sqrt{2}+1)\cos \theta - \frac{1}{\sqrt{2}} = 0 \]

### Step 5: Multiply through by \(\sqrt{2}\) to eliminate the fraction
Multiplying the entire equation by \(\sqrt{2}\) gives:
\[ 2\sqrt{2}\cos^2 \theta - (\sqrt{2}+1)\sqrt{2}\cos \theta - 1 = 0 \]
This simplifies to:
\[ 2\sqrt{2}\cos^2 \theta - (2 + \sqrt{2})\cos \theta - 1 = 0 \]

### Step 6: Use the quadratic formula
This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 2\sqrt{2} \)
- \( b = -(2 + \sqrt{2}) \)
- \( c = -1 \)

Using the quadratic formula \( \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ \cos \theta = \frac{(2 + \sqrt{2}) \pm \sqrt{(2+\sqrt{2})^2 - 4(2\sqrt{2})(-1)}}{2(2\sqrt{2})} \]

### Step 7: Calculate the discriminant
Calculating the discriminant:
\[ (2+\sqrt{2})^2 + 8\sqrt{2} = 4 + 4\sqrt{2} + 2 + 8\sqrt{2} = 6 + 12\sqrt{2} \]

### Step 8: Substitute back into the formula
Substituting back gives:
\[ \cos \theta = \frac{(2 + \sqrt{2}) \pm \sqrt{6 + 12\sqrt{2}}}{4\sqrt{2}} \]

### Step 9: Find specific angles
From the values of \( \cos \theta \), we can find specific angles:
1. If \( \cos \theta = \frac{1}{\sqrt{2}} \), then \( \theta = \frac{\pi}{4} + 2n\pi \) or \( \theta = -\frac{\pi}{4} + 2n\pi \).
2. If \( \cos \theta = \frac{1}{2} \), then \( \theta = \frac{\pi}{3} + 2n\pi \) or \( \theta = -\frac{\pi}{3} + 2n\pi \).

### Final Solution
Thus, the values of \( \theta \) are:
\[ \theta = 2n\pi + \frac{\pi}{4}, \quad 2n\pi - \frac{\pi}{4}, \quad 2n\pi + \frac{\pi}{3}, \quad 2n\pi - \frac{\pi}{3} \]