If `A+B+C=pi`, then value of `|{:(sin(A+B+C),sinB,cosC),(-sinB,0,tanA),(cos(A+B),-tanA,0):}|` is

Options

A. 0
B. 1
C. `2sinB * tanAcosC`
D. `2sinAsinBsinC`

Answer

To solve the determinant given the condition \( A + B + C = \pi \), we will evaluate the determinant step by step.

### Step 1: Write the Determinant
We start with the determinant:
\[ D = \begin{vmatrix} \sin(A+B+C) & \sin B & \cos C \\ -\sin B & 0 & \tan A \\ \cos(A+B) & -\tan A & 0 \end{vmatrix} \]

### Step 2: Substitute \( A + B + C \)
Since \( A + B + C = \pi \), we can substitute this into the determinant:
\[ \sin(A+B+C) = \sin(\pi) = 0 \]
Thus, the determinant becomes:
\[ D = \begin{vmatrix} 0 & \sin B & \cos C \\ -\sin B & 0 & \tan A \\ \cos(A+B) & -\tan A & 0 \end{vmatrix} \]

### Step 3: Expand the Determinant
We can expand the determinant along the first row:
\[ D = 0 \cdot \begin{vmatrix} 0 & \tan A \\ -\tan A & 0 \end{vmatrix} - \sin B \cdot \begin{vmatrix} -\sin B & \tan A \\ \cos(A+B) & 0 \end{vmatrix} + \cos C \cdot \begin{vmatrix} -\sin B & 0 \\ \cos(A+B) & -\tan A \end{vmatrix} \]
Since the first term is multiplied by 0, we focus on the other two terms.

### Step 4: Calculate the Remaining Determinants
1. For the second term:
\[ \begin{vmatrix} -\sin B & \tan A \\ \cos(A+B) & 0 \end{vmatrix} = (-\sin B)(0) - (\tan A)(\cos(A+B)) = -\tan A \cos(A+B) \]

2. For the third term:
\[ \begin{vmatrix} -\sin B & 0 \\ \cos(A+B) & -\tan A \end{vmatrix} = (-\sin B)(-\tan A) - (0)(\cos(A+B)) = \sin B \tan A \]

### Step 5: Substitute Back into the Determinant
Now substituting back:
\[ D = -\sin B (-\tan A \cos(A+B)) + \cos C (\sin B \tan A) \]
This simplifies to:
\[ D = \sin B \tan A \cos(A+B) + \cos C \sin B \tan A \]
Factoring out \(\sin B \tan A\):
\[ D = \sin B \tan A (\cos(A+B) + \cos C) \]

### Step 6: Use the Identity for \(\cos(A+B)\)
Using the identity \(\cos(A+B) = -\cos C\) (since \(A + B + C = \pi\)):
\[ D = \sin B \tan A (-\cos C + \cos C) = \sin B \tan A \cdot 0 = 0 \]

### Conclusion
Thus, the value of the determinant is:
\[ \boxed{0} \]