Which of the following is true ?

Options

A. `sin 765^(@)=-(1)/(sqrt(2))`
B. `cos(-(15pi)/(4))=-1`
C. `tan ((13pi)/3)=(1)/(sqrt(3))`
D. `cosec(-1410^(@))=2`

Explanation

To solve the question "Which of the following is true?", we will evaluate each option step by step.

### Step 1: Evaluate Option A: sin(765°)

1. **Convert 765° to a more manageable angle**:
- \( 765° = 360° + 360° + 45° = 2 \times 360° + 45° \)
- Therefore, \( \sin(765°) = \sin(45°) \).

2. **Calculate sin(45°)**:
- \( \sin(45°) = \frac{1}{\sqrt{2}} \).

3. **Conclusion for Option A**:
- \( \sin(765°) = \frac{1}{\sqrt{2}} \), which means Option A is **false**.

### Step 2: Evaluate Option B: cos(-15π/4)

1. **Use the property of cosine**:
- \( \cos(-\theta) = \cos(\theta) \), so \( \cos(-15\pi/4) = \cos(15\pi/4) \).

2. **Convert 15π/4 to a more manageable angle**:
- \( 15\pi/4 = 4\pi + \pi/4 \).
- Since \( 4\pi \) is a multiple of \( 2\pi \), we can simplify:
- \( \cos(15\pi/4) = \cos(\pi/4) \).

3. **Calculate cos(π/4)**:
- \( \cos(\pi/4) = \frac{1}{\sqrt{2}} \).

4. **Conclusion for Option B**:
- \( \cos(-15\pi/4) = \frac{1}{\sqrt{2}} \), so Option B is **false**.

### Step 3: Evaluate Option C: tan(13π/3)

1. **Convert 13π/3 to a more manageable angle**:
- \( 13\pi/3 = 12\pi/3 + \pi/3 = 4\pi + \pi/3 \).
- Since \( 4\pi \) is a multiple of \( 2\pi \), we can simplify:
- \( \tan(13\pi/3) = \tan(\pi/3) \).

2. **Calculate tan(π/3)**:
- \( \tan(\pi/3) = \sqrt{3} \).

3. **Conclusion for Option C**:
- \( \tan(13\pi/3) = \sqrt{3} \), so Option C is **false**.

### Step 4: Evaluate Option D: cosec(-1410°)

1. **Use the property of sine**:
- \( \csc(-\theta) = -\csc(\theta) \), so we need to find \( \sin(1410°) \).

2. **Convert 1410° to a more manageable angle**:
- \( 1410° = 8 \times 180° - 30° = 1440° - 30° \).
- Hence, \( \sin(1410°) = \sin(-30°) \).

3. **Calculate sin(-30°)**:
- \( \sin(-30°) = -\frac{1}{2} \).

4. **Calculate cosec(-1410°)**:
- \( \csc(-1410°) = -\frac{1}{\sin(1410°)} = -\frac{1}{-\frac{1}{2}} = 2 \).

5. **Conclusion for Option D**:
- \( \csc(-1410°) = 2 \), so Option D is **true**.

### Final Conclusion:
The correct option is **D**: \( \csc(-1410°) = 2 \).

---