ABCD is square. K and L are two points on AB such that AO = AK and BO = BL and `angleLOK=theta`.
Find the value of `tantheta`.

Answer

To solve the problem, we need to find the value of \( \tan \theta \) given that \( ABCD \) is a square, \( K \) and \( L \) are points on side \( AB \), and \( AO = AK \) and \( BO = BL \) with \( \angle LOK = \theta \).

### Step-by-Step Solution:

1. **Define the Square**:
Let the square \( ABCD \) have a side length of \( 2 \). Therefore, the coordinates of the points are:
- \( A(0, 2) \)
- \( B(2, 2) \)
- \( C(2, 0) \)
- \( D(0, 0) \)

2. **Identify Points K and L**:
Since \( AO = AK \) and \( BO = BL \), let \( AO = AK = x \) and \( BO = BL = y \). Thus, the coordinates of points \( K \) and \( L \) can be defined as:
- \( K(0, 2 - x) \)
- \( L(2, 2 - y) \)

3. **Find the Coordinates of O**:
The center \( O \) of the square \( ABCD \) is the midpoint of the diagonal \( AC \). Therefore, the coordinates of \( O \) are:
\[ O\left(1, 1\right) \]

4. **Calculate the Lengths**:
The lengths \( OK \) and \( OL \) can be calculated using the distance formula:
- \( OK = \sqrt{(1 - 0)^2 + (1 - (2 - x))^2} = \sqrt{1 + (x - 1)^2} \)
- \( OL = \sqrt{(1 - 2)^2 + (1 - (2 - y))^2} = \sqrt{1 + (y - 1)^2} \)

5. **Use the Angle Definition**:
From the triangle \( LOK \), we know that:
\[ \tan \theta = \frac{LM}{OL} \]
where \( LM \) is the length of the line segment between points \( L \) and \( K \).

6. **Calculate LM**:
The length \( LM \) can be calculated as:
\[ LM = \sqrt{(2 - 0)^2 + ((2 - y) - (2 - x))^2} = \sqrt{4 + (x - y)^2} \]

7. **Substituting into the Tangent Formula**:
We can now substitute the values into the tangent formula:
\[ \tan \theta = \frac{\sqrt{4 + (x - y)^2}}{\sqrt{1 + (x - 1)^2}} \]

8. **Simplifying the Expression**:
To find \( \tan \theta \), we can simplify this expression further based on the values of \( x \) and \( y \). If we assume \( x = y \), then:
\[ \tan \theta = \frac{\sqrt{4}}{\sqrt{1}} = 2 \]

### Conclusion:
Thus, the value of \( \tan \theta \) is \( 2 \).