Consider the following for triangle ABC :
1. `sin((B+C)/(2))=cos((A)/(2))`
2. `tan((B+C)/(2))=cot((A)/(2))`
3. sin (B + C) = cos A
4. tan(B + C) = -cot A
Which of the above are correct ?

Options

A. 1 and 3
B. 1 and 2
C. 1 and 4
D. 2 and 2

Explanation

To determine which of the statements regarding triangle ABC are correct, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement 1:** \( \sin\left(\frac{B+C}{2}\right) = \cos\left(\frac{A}{2}\right) \)

From the triangle property, we know that:
\[ A + B + C = 180^\circ \quad \Rightarrow \quad B + C = 180^\circ - A \]
Thus, we can express \( B + C \) in terms of \( A \):
\[ \frac{B+C}{2} = \frac{180^\circ - A}{2} = 90^\circ - \frac{A}{2} \]
Now, using the sine function:
\[ \sin\left(\frac{B+C}{2}\right) = \sin\left(90^\circ - \frac{A}{2}\right) = \cos\left(\frac{A}{2}\right) \]
This shows that Statement 1 is **correct**.

### Step 2: Analyze the second statement
**Statement 2:** \( \tan\left(\frac{B+C}{2}\right) = \cot\left(\frac{A}{2}\right) \)

Using the same expression for \( \frac{B+C}{2} \):
\[ \tan\left(\frac{B+C}{2}\right) = \tan\left(90^\circ - \frac{A}{2}\right) = \cot\left(\frac{A}{2}\right) \]
This shows that Statement 2 is also **correct**.

### Step 3: Analyze the third statement
**Statement 3:** \( \sin(B+C) = \cos A \)

We know that:
\[ B + C = 180^\circ - A \quad \Rightarrow \quad \sin(B+C) = \sin(180^\circ - A) = \sin A \]
Since \( \sin A \) is not equal to \( \cos A \) in general, Statement 3 is **incorrect**.

### Step 4: Analyze the fourth statement
**Statement 4:** \( \tan(B+C) = -\cot A \)

Using the same expression for \( B + C \):
\[ \tan(B+C) = \tan(180^\circ - A) = -\tan A \]
And since \( \cot A = \frac{1}{\tan A} \), we have:
\[ -\cot A = -\frac{1}{\tan A} \]
Thus, Statement 4 is also **incorrect**.

### Conclusion
The correct statements are:
- Statement 1: Correct
- Statement 2: Correct
- Statement 3: Incorrect
- Statement 4: Incorrect

Therefore, the answer is that **only statements 1 and 2 are correct**.