Consider the following two statements :
Statement p : The value of can be derived by taking `theta = 240^(@)` in the equation `2 sin.(theta)/(2) = sqrt(1+ sin theta) - sqrt(1- sin theta)`
Statement q : The angles A, B, C and D of any quadrilateral ABCD satisfy the equation `cos((1)/(2)(A+C)) + cos((1)/(2)(B+D)) = 0`
Then the truth value of p and q are respectively :

Options

A. T, T
B. F, F
C. F, T
D. T, F

Answer

To analyze the truth values of the statements p and q, we will go through each statement step by step.

### Step 1: Evaluate Statement p

**Statement p**: The value of can be derived by taking `theta = 240°` in the equation

\[ \frac{2 \sin(\theta)}{2} = \sqrt{1 + \sin(\theta)} - \sqrt{1 - \sin(\theta)} \]

1. Substitute \(\theta = 240°\) into the equation:
\[ 2 \sin(240°) = \sqrt{1 + \sin(240°)} - \sqrt{1 - \sin(240°)} \]

2. Calculate \(\sin(240°)\):
\[ \sin(240°) = -\frac{\sqrt{3}}{2} \]
Thus, we have:
\[ 2 \left(-\frac{\sqrt{3}}{2}\right) = \sqrt{1 - \frac{\sqrt{3}}{2}} - \sqrt{1 + \frac{\sqrt{3}}{2}} \]
Simplifying gives:
\[ -\sqrt{3} = \sqrt{1 - \frac{\sqrt{3}}{2}} - \sqrt{1 + \frac{\sqrt{3}}{2}} \]

3. Calculate the right-hand side:
- Calculate \(1 - \frac{\sqrt{3}}{2}\):
\[ 1 - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2} \]
- Calculate \(1 + \frac{\sqrt{3}}{2}\):
\[ 1 + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2} \]

4. Substitute these values back into the equation:
\[ -\sqrt{3} = \sqrt{\frac{2 - \sqrt{3}}{2}} - \sqrt{\frac{2 + \sqrt{3}}{2}} \]

5. This equation does not hold true, hence **Statement p is false**.

### Step 2: Evaluate Statement q

**Statement q**: The angles A, B, C, and D of any quadrilateral ABCD satisfy the equation

\[ \cos\left(\frac{1}{2}(A + C)\right) + \cos\left(\frac{1}{2}(B + D)\right) = 0 \]

1. From the properties of a quadrilateral, we know that:
\[ A + B + C + D = 360° \]
Thus, we can express \(A + C\) and \(B + D\) as:
\[ A + C = 360° - (B + D) \]

2. Therefore, we can write:
\[ \frac{1}{2}(A + C) + \frac{1}{2}(B + D) = 180° \]

3. This implies:
\[ \cos\left(\frac{1}{2}(A + C)\right) = -\cos\left(\frac{1}{2}(B + D)\right) \]

4. Hence, it follows that:
\[ \cos\left(\frac{1}{2}(A + C)\right) + \cos\left(\frac{1}{2}(B + D)\right) = 0 \]

5. Therefore, **Statement q is true**.

### Conclusion

- **Statement p** is **false**.
- **Statement q** is **true**.

Thus, the truth values of p and q are respectively: **false** and **true**.