The number of values of x in `(0, pi)` satisfying the equation
`(sqrt(3) "sin" x + "cos" x) ^(sqrt(sqrt(3)"sin" 2x -"cos" 2x+ 2)) = 4`, is

Options

A. 0
B. 1
C. 2
D. none of these

Explanation

To solve the equation

\[ (\sqrt{3} \sin x + \cos x)^{\sqrt{\sqrt{3} \sin 2x - \cos 2x + 2}} = 4 \]

we will follow these steps:

### Step 1: Simplify the exponent
The exponent is given as

\[ \sqrt{3} \sin 2x - \cos 2x + 2 \]

We can rewrite \(\sin 2x\) and \(\cos 2x\) using the double angle identities:

\[ \sin 2x = 2 \sin x \cos x \quad \text{and} \quad \cos 2x = \cos^2 x - \sin^2 x \]

Substituting these into the exponent gives:

\[ \sqrt{3}(2 \sin x \cos x) - (\cos^2 x - \sin^2 x) + 2 \]

### Step 2: Further simplify the exponent
This can be rearranged as:

\[ 2\sqrt{3} \sin x \cos x - (\cos^2 x - \sin^2 x) + 2 \]

Now, recognizing that \(\cos^2 x + \sin^2 x = 1\), we can simplify further:

\[ = 2\sqrt{3} \sin x \cos x + 2 - \cos^2 x + \sin^2 x \]

### Step 3: Set the equation for simplification
Now we can express the original equation as:

\[ (\sqrt{3} \sin x + \cos x)^{\sqrt{3} \sin 2x - \cos 2x + 2} = 4 \]

Since \(4 = 2^2\), we can equate the bases and the powers:

\[ \sqrt{3} \sin x + \cos x = 2 \]

### Step 4: Solve for \(x\)
Rearranging gives:

\[ \sqrt{3} \sin x + \cos x = 2 \]

To solve for \(x\), we can express this in terms of sine and cosine:

\[ \sqrt{3} \sin x + \cos x = 2 \]

This equation can be interpreted geometrically. The maximum value of \(\sqrt{3} \sin x + \cos x\) occurs when the angle is at its maximum, which is:

\[ \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2 \]

This maximum occurs when:

\[ \tan \theta = \frac{1}{\sqrt{3}} \implies \theta = \frac{\pi}{6} \]

Thus, we have:

\[ \sqrt{3} \sin x + \cos x = 2 \implies \text{only when } x = \frac{\pi}{6} + n\pi \]

### Step 5: Check the interval
Since we are looking for solutions in the interval \( (0, \pi) \):

1. \(x = \frac{\pi}{6}\) is valid.
2. Check if there are any other solutions in \( (0, \pi) \).

### Conclusion
The only solution in the interval \( (0, \pi) \) is \( x = \frac{\pi}{3} \).

Thus, the number of values of \(x\) satisfying the equation is:

\[ \boxed{1} \]