Three expressions are given below :
`Q_(1)=sin(A+B)+sin(B+C)+sin(C+A)`
`Q_(2)=cos(A-B)+cos(B-C)+cos(C-A)`
`Q_(3)=sinA(cosB+cosC)+sinB(cosC+cosA)+sinC(cosA+cosB)`
Which one the following is correct ?

Options

A. `Q_(1)=Q_(2)`
B. `Q_(2)=Q_(3)`
C. `Q_(1)=Q_(3)`
D. All the expressions are different

Explanation

To solve the problem, we need to analyze the three given expressions \( Q_1 \), \( Q_2 \), and \( Q_3 \) and determine which one is correct.

### Step 1: Analyze \( Q_1 \)

The first expression is:
\[ Q_1 = \sin(A + B) + \sin(B + C) + \sin(C + A) \]

Using the sine addition formula, we can expand each term:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
\[ \sin(B + C) = \sin B \cos C + \cos B \sin C \]
\[ \sin(C + A) = \sin C \cos A + \cos C \sin A \]

Combining these, we have:
\[ Q_1 = (\sin A \cos B + \cos A \sin B) + (\sin B \cos C + \cos B \sin C) + (\sin C \cos A + \cos C \sin A) \]

### Step 2: Analyze \( Q_2 \)

The second expression is:
\[ Q_2 = \cos(A - B) + \cos(B - C) + \cos(C - A) \]

Using the cosine subtraction formula:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
\[ \cos(B - C) = \cos B \cos C + \sin B \sin C \]
\[ \cos(C - A) = \cos C \cos A + \sin C \sin A \]

Combining these, we have:
\[ Q_2 = (\cos A \cos B + \sin A \sin B) + (\cos B \cos C + \sin B \sin C) + (\cos C \cos A + \sin C \sin A) \]

### Step 3: Analyze \( Q_3 \)

The third expression is:
\[ Q_3 = \sin A (\cos B + \cos C) + \sin B (\cos C + \cos A) + \sin C (\cos A + \cos B) \]

This expression is already in a simplified form, but we can rearrange it:
\[ Q_3 = \sin A \cos B + \sin A \cos C + \sin B \cos C + \sin B \cos A + \sin C \cos A + \sin C \cos B \]

### Step 4: Compare the Expressions

Now, we can compare \( Q_1 \) and \( Q_3 \):
- From the expansions, we see that both \( Q_1 \) and \( Q_3 \) consist of similar terms:
\[ Q_1 = \sin A \cos B + \sin B \cos C + \sin C \cos A + \cos A \sin B + \cos B \sin C + \cos C \sin A \]
\[ Q_3 = \sin A \cos B + \sin A \cos C + \sin B \cos C + \sin B \cos A + \sin C \cos A + \sin C \cos B \]

Thus, we can conclude that:
\[ Q_1 = Q_3 \]

### Conclusion

Since \( Q_1 \) is equal to \( Q_3 \), we find that:
- \( Q_1 = Q_3 \)
- \( Q_2 \) does not equal \( Q_1 \) or \( Q_3 \).

Thus, the correct answer is that \( Q_1 \) and \( Q_3 \) are equal.

### Final Answer
The correct option is \( Q_1 = Q_3 \).