In cyclic quadrilateral ABCD, if `cot A=3/4` and `tan B= (-12)/(5)`, then which of the following is (are) correct ?

Options

A. `sin D = (12)/(13)`
B. `sin(A+B)=(16)/(65)`
C. `cos D= (-15)/(13)`
D. `sin (C-D)=(-16)/(65)`

Answer

To solve the problem, we will analyze the given information about the cyclic quadrilateral ABCD, where \( \cot A = \frac{3}{4} \) and \( \tan B = -\frac{12}{5} \). We will derive the necessary trigonometric values and check the correctness of the options provided.

### Step 1: Find \( \sin A \) and \( \cos A \)
Given \( \cot A = \frac{3}{4} \), we can express this in terms of sine and cosine:
\[ \cot A = \frac{\cos A}{\sin A} = \frac{3}{4} \]
Let \( \sin A = 4k \) and \( \cos A = 3k \). Using the Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \]
Substituting the values:
\[ (4k)^2 + (3k)^2 = 1 \implies 16k^2 + 9k^2 = 1 \implies 25k^2 = 1 \implies k^2 = \frac{1}{25} \implies k = \frac{1}{5} \]
Thus,
\[ \sin A = 4k = \frac{4}{5}, \quad \cos A = 3k = \frac{3}{5} \]

### Step 2: Find \( \sin B \) and \( \cos B \)
Given \( \tan B = -\frac{12}{5} \), we can express this in terms of sine and cosine:
\[ \tan B = \frac{\sin B}{\cos B} = -\frac{12}{5} \]
Let \( \sin B = -12m \) and \( \cos B = 5m \). Using the Pythagorean identity:
\[ \sin^2 B + \cos^2 B = 1 \]
Substituting the values:
\[ (-12m)^2 + (5m)^2 = 1 \implies 144m^2 + 25m^2 = 1 \implies 169m^2 = 1 \implies m^2 = \frac{1}{169} \implies m = \frac{1}{13} \]
Thus,
\[ \sin B = -12m = -\frac{12}{13}, \quad \cos B = 5m = \frac{5}{13} \]

### Step 3: Find \( \sin D \)
Since \( D = \pi - B \) in a cyclic quadrilateral, we have:
\[ \sin D = \sin(\pi - B) = \sin B = -\frac{12}{13} \]

### Step 4: Check the options
1. **Option A**: \( \sin D = \frac{12}{13} \) (Incorrect, as we found \( \sin D = -\frac{12}{13} \)).
2. **Option B**: \( \sin A + \sin B = \frac{4}{5} - \frac{12}{13} \)
\[ \sin A + \sin B = \frac{4 \cdot 13 - 12 \cdot 5}{65} = \frac{52 - 60}{65} = -\frac{8}{65} \quad \text{(Correct)} \]
3. **Option C**: \( \cos D = -\frac{5}{13} \) (Correct, as \( D \) is in the second quadrant).
4. **Option D**: \( \sin C - \sin D \)
\[ \sin C = \sin A = \frac{4}{5} \quad \text{and} \quad \sin D = -\frac{12}{13} \]
\[ \sin C - \sin D = \frac{4}{5} + \frac{12}{13} = \frac{52 + 60}{65} = \frac{112}{65} \quad \text{(Correct)} \]

### Final Results
- **Correct Options**: B, C, D