In `Delta ABC` if BC is unity, `sin ""A/2=x _(1), sin ""B/2 =x _(2), cos "" A/2 =x _(3) and cos "" B/2 =x _(4)` with `((x _(1))/(x _(2)))^(2007) - ((x _(3))/(x_(4)))^(2006) =0,` then the length of AC is `"________"`

Answer

To solve the problem, we start with the given information about triangle ABC, where BC = 1. We are also given the following:

- \( \sin \frac{A}{2} = x_1 \)
- \( \sin \frac{B}{2} = x_2 \)
- \( \cos \frac{A}{2} = x_3 \)
- \( \cos \frac{B}{2} = x_4 \)

The equation we need to analyze is:

\[ \left( \frac{x_1}{x_2} \right)^{2007} - \left( \frac{x_3}{x_4} \right)^{2006} = 0 \]

### Step 1: Analyze the given equation

From the equation, we can rewrite it as:

\[ \left( \frac{x_1}{x_2} \right)^{2007} = \left( \frac{x_3}{x_4} \right)^{2006} \]

### Step 2: Consider the implications of the equation

Taking the 2007th root on both sides gives:

\[ \frac{x_1}{x_2} = \left( \frac{x_3}{x_4} \right)^{\frac{2006}{2007}} \]

### Step 3: Use properties of sine and cosine

Since \( \sin \frac{A}{2} \) and \( \sin \frac{B}{2} \) are positive in the interval \( (0, \frac{\pi}{2}) \), we know that:

- \( \sin \frac{A}{2} \) increases as \( A \) increases.
- \( \cos \frac{A}{2} \) decreases as \( A \) increases.

### Step 4: Analyze the cases

1. **Case 1**: If \( \frac{A}{2} < \frac{B}{2} \) (which implies \( A < B \)):
- This leads to \( x_1 < x_2 \) (since sine is increasing).
- Also, \( x_3 > x_4 \) (since cosine is decreasing).
- This contradicts the equality derived from the equation.

2. **Case 2**: If \( \frac{A}{2} > \frac{B}{2} \) (which implies \( A > B \)):
- This leads to \( x_1 > x_2 \).
- Also, \( x_3 < x_4 \).
- This again contradicts the equality derived from the equation.

### Step 5: Conclusion from cases

The only way for the equation to hold true is if:

\[ \frac{A}{2} = \frac{B}{2} \implies A = B \]

This means triangle ABC is isosceles with \( AC = AB \).

### Step 6: Determine the length of AC

Since \( BC = 1 \) and \( A = B \), by the properties of isosceles triangles, we can conclude that:

- The sides opposite equal angles are equal.
- Therefore, \( AC = AB \).

Using the Law of Cosines or basic properties of isosceles triangles, we find that:

\[ AC = 1 \]

Thus, the length of AC is:

\[ \boxed{1} \]