The value of x satisfying the equation `x=sqrt(2+sqrt(2-sqrt(2+x)))` is

Options

A. `2 cos 10^(@)`
B. `2 cos 20^(@)`
C. `2 cos 40^(@)`
D. `2 cos 80^(@)`

Explanation

To solve the equation \( x = \sqrt{2 + \sqrt{2 - \sqrt{2 + x}}} \), we will follow a systematic approach:

### Step 1: Substitute \( x \)
Let's start by substituting \( x \) with \( 2 \cos \theta \) for some angle \( \theta \):
\[ x = 2 \cos \theta \]

### Step 2: Rewrite the equation
Substituting \( x \) in the original equation gives:
\[ 2 \cos \theta = \sqrt{2 + \sqrt{2 - \sqrt{2 + 2 \cos \theta}}} \]

### Step 3: Simplify the right-hand side
We simplify the right-hand side:
\[ \sqrt{2 - \sqrt{2 + 2 \cos \theta}} = \sqrt{2 - \sqrt{2(1 + \cos \theta)}} \]
Using the identity \( 1 + \cos \theta = 2 \cos^2(\theta/2) \):
\[ \sqrt{2(1 + \cos \theta)} = \sqrt{2 \cdot 2 \cos^2(\theta/2)} = 2 \cos(\theta/2) \]
Thus:
\[ \sqrt{2 - 2 \cos(\theta/2)} = \sqrt{2(1 - \cos(\theta/2))} = \sqrt{2 \cdot 2 \sin^2(\theta/4)} = 2 \sin(\theta/4) \]

### Step 4: Substitute back
Now substituting back, we have:
\[ 2 \cos \theta = \sqrt{2 + 2 \sin(\theta/4)} \]

### Step 5: Square both sides
Squaring both sides gives:
\[ 4 \cos^2 \theta = 2 + 2 \sin(\theta/4) \]
This simplifies to:
\[ 2 \cos^2 \theta - 1 = \sin(\theta/4) \]

### Step 6: Use trigonometric identities
Using the identity \( \cos 2\theta = 2 \cos^2 \theta - 1 \):
\[ \cos 2\theta = \sin(\theta/4) \]

### Step 7: Solve for \( \theta \)
To solve for \( \theta \), we can use the identity:
\[ \cos 2\theta = \sin\left(\frac{\theta}{4}\right) \]
This implies:
\[ 2\theta = 90^\circ - \frac{\theta}{4} + k \cdot 360^\circ \quad \text{(for some integer \( k \))} \]
Solving this gives:
\[ 8\theta + \theta = 360^\circ k + 90^\circ \]
\[ 9\theta = 360^\circ k + 90^\circ \]
\[ \theta = 40^\circ + 40^\circ k \]

### Step 8: Find \( x \)
Taking \( k = 0 \):
\[ \theta = 40^\circ \]
Thus:
\[ x = 2 \cos(40^\circ) \]

### Final Answer
The value of \( x \) satisfying the equation is:
\[ \boxed{2 \cos 40^\circ} \]