`"cos" 2x =(sqrt(2) + 1) ("cos"x- (1)/(sqrt(2))),"cos" x ne (1)/(2) rArr x in `

Options

A. `{2n pi +-(pi)/(3): n in Z}`
B. `{2npi +-(pi)/(6): n inZ}`
C. `{2npi + - (pi)/(2): n inZ}`
D. `{2npi +- (pi)/(4): n inZ}`

Explanation

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

### Step 1: Rewrite \( \cos 2x \)
We know that \( \cos 2x \) can be expressed as:
\[ \cos 2x = 2 \cos^2 x - 1 \]
So, we can rewrite the equation as:
\[ 2 \cos^2 x - 1 = (\sqrt{2} + 1) \left( \cos x - \frac{1}{\sqrt{2}} \right) \]

### Step 2: Expand the right-hand side
Now, we will expand the right-hand side:
\[ (\sqrt{2} + 1) \left( \cos x - \frac{1}{\sqrt{2}} \right) = (\sqrt{2} + 1) \cos x - (\sqrt{2} + 1) \cdot \frac{1}{\sqrt{2}} \]
This simplifies to:
\[ (\sqrt{2} + 1) \cos x - (1 + \frac{1}{\sqrt{2}}) \]

### Step 3: Set the equation to zero
Now, we can set the equation to zero:
\[ 2 \cos^2 x - 1 - \left( (\sqrt{2} + 1) \cos x - (1 + \frac{1}{\sqrt{2}}) \right) = 0 \]
Rearranging gives us:
\[ 2 \cos^2 x - (\sqrt{2} + 1) \cos x + \left( 1 + \frac{1}{\sqrt{2}} - 1 \right) = 0 \]
This simplifies to:
\[ 2 \cos^2 x - (\sqrt{2} + 1) \cos x + \frac{1}{\sqrt{2}} = 0 \]

### Step 4: Solve the quadratic equation
Let \( y = \cos x \). The equation becomes:
\[ 2y^2 - (\sqrt{2} + 1)y + \frac{1}{\sqrt{2}} = 0 \]
We can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ y = \frac{(\sqrt{2} + 1) \pm \sqrt{(\sqrt{2} + 1)^2 - 4 \cdot 2 \cdot \frac{1}{\sqrt{2}}}}{2 \cdot 2} \]

### Step 5: Calculate the discriminant
Calculating the discriminant:
\[ (\sqrt{2} + 1)^2 - 4 \cdot 2 \cdot \frac{1}{\sqrt{2}} = 2 + 2\sqrt{2} + 1 - \frac{8}{\sqrt{2}} = 3 + 2\sqrt{2} - 4\sqrt{2} = 3 - 2\sqrt{2} \]

### Step 6: Find the roots
Now substituting back into the quadratic formula:
\[ y = \frac{(\sqrt{2} + 1) \pm \sqrt{3 - 2\sqrt{2}}}{4} \]

### Step 7: Find \( x \)
Since we have \( y = \cos x \), we can find \( x \) using:
\[ x = \cos^{-1}\left(\frac{(\sqrt{2} + 1) \pm \sqrt{3 - 2\sqrt{2}}}{4}\right) \]
However, we must ensure that \( \cos x \neq \frac{1}{2} \).

### Step 8: General solution
The general solution for \( x \) based on the values of \( \cos x \) will be:
\[ x = 2n\pi \pm \theta \quad \text{where } \theta = \cos^{-1}\left(\frac{(\sqrt{2} + 1) \pm \sqrt{3 - 2\sqrt{2}}}{4}\right) \]