If in a triangle ABC, `theta` is the angle determined by `cos theta=(a-b)//c`, then

Options

A. `((a+b)sin theta)/(2sqrt(ab))=cos(A-B)/(2)`
B. `((a+b)sin theta)/(2sqrt(ab))=cos(A+B)/(2)`
C. `(c sin theta)/(2sqrt(ab))=cos(A-B)/(2)`
D. `(c sin theta)/(2sqrt(ab))=cos(A+B)/(2)`

Explanation

To solve the problem, we need to analyze the given equation \( \cos \theta = \frac{A - B}{C} \) in the context of triangle \( ABC \).

### Step-by-step Solution:

1. **Understanding the Given Equation**:
We start with the equation:
\[ \cos \theta = \frac{A - B}{C} \]
where \( A \), \( B \), and \( C \) are the lengths of the sides opposite to angles \( A \), \( B \), and \( C \) respectively.

2. **Using the Law of Sines**:
According to the Law of Sines:
\[ \frac{A}{\sin A} = \frac{B}{\sin B} = \frac{C}{\sin C} = k \]
where \( k \) is a constant.

3. **Expressing Sides in Terms of Angles**:
From the Law of Sines, we can express \( A \), \( B \), and \( C \) as:
\[ A = k \sin A, \quad B = k \sin B, \quad C = k \sin C \]
Substituting these into the equation for \( \cos \theta \):
\[ \cos \theta = \frac{k \sin A - k \sin B}{k \sin C} = \frac{\sin A - \sin B}{\sin C} \]

4. **Using the Sine Difference Formula**:
We can apply the sine difference formula:
\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
Therefore, we can rewrite \( \cos \theta \):
\[ \cos \theta = \frac{2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)}{\sin C} \]

5. **Using the Sine of Angle C**:
Using the angle sum property \( A + B + C = 180^\circ \) or \( \pi \) radians:
\[ C = \pi - (A + B) \implies \sin C = \sin(A + B) \]
We can express \( \sin(A + B) \) using the sine addition formula:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]

6. **Substituting Back**:
Now substituting back into our expression for \( \cos \theta \):
\[ \cos \theta = \frac{2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)}{\sin(A + B)} \]

7. **Finding Sine of Theta**:
We can find \( \sin \theta \) using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]

8. **Finalizing the Options**:
We can now check the multiple-choice options provided in the question against our derived expressions for \( \sin \theta \) and \( \cos \theta \) to determine which options are correct.