Solve for `theta`: `cos2theta = (sqrt2+1)(costheta - (1)/(sqrt2))`.

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: Use the double angle identity for cosine
We know that:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
So we can rewrite the equation as:
\[ 2\cos^2 \theta - 1 = (\sqrt{2} + 1)\left( \cos \theta - \frac{1}{\sqrt{2}} \right) \]

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

### Step 3: Rearrange the equation
Rearranging gives us:
\[ 2\cos^2 \theta - (\sqrt{2} + 1)\cos \theta + \left(1 + \frac{1}{\sqrt{2}} - 1\right) = 0 \]
This simplifies to:
\[ 2\cos^2 \theta - (\sqrt{2} + 1)\cos \theta + \frac{1}{\sqrt{2}} = 0 \]

### Step 4: Form a quadratic equation
Let \( x = \cos \theta \). The equation becomes:
\[ 2x^2 - (\sqrt{2} + 1)x + \frac{1}{\sqrt{2}} = 0 \]

### Step 5: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 2 \), \( b = -(\sqrt{2} + 1) \), and \( c = \frac{1}{\sqrt{2}} \).

Calculating the discriminant:
\[ b^2 - 4ac = (\sqrt{2} + 1)^2 - 4 \cdot 2 \cdot \frac{1}{\sqrt{2}} \]
Calculating \( (\sqrt{2} + 1)^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \) and \( 4 \cdot 2 \cdot \frac{1}{\sqrt{2}} = 4\sqrt{2} \):
\[ b^2 - 4ac = (3 + 2\sqrt{2}) - 4\sqrt{2} = 3 - 2\sqrt{2} \]

### Step 6: Solve for \( x \)
Now we can substitute back into the quadratic formula:
\[ x = \frac{\sqrt{2} + 1 \pm \sqrt{3 - 2\sqrt{2}}}{4} \]

### Step 7: Find \( \theta \)
After calculating \( x \), we can find \( \theta \) using:
\[ \theta = \cos^{-1}(x) \]
We will find the angles corresponding to the values of \( x \).