The number of solutions of equation `|(1,1,1), (1,1+sin theta,1), (1,1, 1+cot theta)|=0` in `theta in [0, 2 pi]` is equal to

Options

A. 2
B. 3
C. 4
D. 5

Explanation

To solve the equation given by the determinant

\[ \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 & 1 & 1 + \cot \theta \end{vmatrix} = 0 \]

we will follow these steps:

### Step 1: Compute the Determinant

We can compute the determinant using the formula for a 3x3 determinant:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our determinant, we have:

\[ D = 1 \cdot \left( (1 + \sin \theta)(1 + \cot \theta) - 1 \cdot 1 \right) - 1 \cdot \left( 1 \cdot (1 + \cot \theta) - 1 \cdot 1 \right) + 1 \cdot \left( 1 \cdot 1 - 1 \cdot (1 + \sin \theta) \right) \]

Calculating each term:

1. **First term**:
\[ (1 + \sin \theta)(1 + \cot \theta) - 1 = 1 + \sin \theta + \cot \theta + \sin \theta \cot \theta - 1 = \sin \theta + \cot \theta + \sin \theta \cot \theta \]

2. **Second term**:
\[ 1 + \cot \theta - 1 = \cot \theta \]

3. **Third term**:
\[ 1 - (1 + \sin \theta) = -\sin \theta \]

Putting it all together, we have:

\[ D = \sin \theta + \cot \theta + \sin \theta \cot \theta - \cot \theta - \sin \theta = \sin \theta \cot \theta = 0 \]

### Step 2: Solve the Equation

The equation simplifies to:

\[ \sin \theta \cot \theta = 0 \]

This can be rewritten as:

\[ \sin \theta \cdot \frac{\cos \theta}{\sin \theta} = 0 \]

This implies:

\[ \cos \theta = 0 \]

### Step 3: Find the Solutions in the Interval [0, 2π]

The solutions for \(\cos \theta = 0\) within the interval \([0, 2\pi]\) are:

\[ \theta = \frac{\pi}{2}, \frac{3\pi}{2} \]

### Conclusion

Thus, the number of solutions of the equation in the interval \([0, 2\pi]\) is **2**.

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