If either of the two `P`, `Q` and `R` are equal and `P+Q+R=180^(@)`, then the value of `|{:(1, 1+sinP,sinP(1+sinP)),(1,1+sinQ,sinQ(1+sinQ)),(1,1+sinR,sinR(1+sinR)):}|` is

Options

A. `0`
B. `1`
C. `sin(P+Q+R)`
D. `sinPsinQsinR`

Explanation

To solve the problem, we need to find the value of the determinant given the angles \( P \), \( Q \), and \( R \) such that \( P + Q + R = 180^\circ \) and at least two of the angles are equal.

The determinant we need to evaluate is:

\[ D = \begin{vmatrix} 1 & 1 + \sin P & \sin P(1 + \sin P) \\ 1 & 1 + \sin Q & \sin Q(1 + \sin Q) \\ 1 & 1 + \sin R & \sin R(1 + \sin R) \end{vmatrix} \]

### Step 1: Subtract Column 1 from Column 2

We will perform the operation \( C_2 \to C_2 - C_1 \):

\[ D = \begin{vmatrix} 1 & \sin P & \sin P(1 + \sin P) \\ 1 & \sin Q & \sin Q(1 + \sin Q) \\ 1 & \sin R & \sin R(1 + \sin R) \end{vmatrix} \]

### Step 2: Subtract Column 2 from Column 3

Next, we perform the operation \( C_3 \to C_3 - C_2 \):

\[ D = \begin{vmatrix} 1 & \sin P & \sin P^2 \\ 1 & \sin Q & \sin Q^2 \\ 1 & \sin R & \sin R^2 \end{vmatrix} \]

### Step 3: Apply Row Operations

Now, we will apply row operations to simplify the determinant. We can subtract Row 1 from Rows 2 and 3:

\[ D = \begin{vmatrix} 1 & \sin P & \sin P^2 \\ 0 & \sin Q - \sin P & \sin Q^2 - \sin P^2 \\ 0 & \sin R - \sin P & \sin R^2 - \sin P^2 \end{vmatrix} \]

### Step 4: Factor the Determinant

Now we can factor the determinant:

\[ D = \begin{vmatrix} 1 & \sin P & \sin P^2 \\ 0 & \sin Q - \sin P & (\sin Q - \sin P)(\sin Q + \sin P) \\ 0 & \sin R - \sin P & (\sin R - \sin P)(\sin R + \sin P) \end{vmatrix} \]

### Step 5: Expand the Determinant

The determinant simplifies to:

\[ D = (\sin Q - \sin P)(\sin R - \sin P) \cdot \begin{vmatrix} 1 & \sin P \\ 1 & \sin R + \sin P \end{vmatrix} \]

### Step 6: Calculate the 2x2 Determinant

Calculating the 2x2 determinant:

\[ \begin{vmatrix} 1 & \sin P \\ 1 & \sin R + \sin P \end{vmatrix} = 1(\sin R + \sin P) - 1(\sin P) = \sin R \]

### Step 7: Final Expression for the Determinant

Thus, we have:

\[ D = (\sin Q - \sin P)(\sin R - \sin P) \cdot \sin R \]

### Step 8: Evaluate the Determinant

Since we know that at least two angles among \( P \), \( Q \), and \( R \) are equal, this means that either \( \sin Q = \sin P \) or \( \sin R = \sin P \). Therefore, at least one of the factors in the product will be zero, leading to:

\[ D = 0 \]

### Conclusion

The value of the determinant is \( 0 \).