If `cot A=[(sin B)/((1- cos B))]`, then what is the value of `cot 2A`?

Options

A. `cot (B/2)`
B. `cot 2B`
C. `cot B`
D. `tan B`

Answer

To solve the problem, we start with the given equation:

\[ \cot A = \frac{\sin B}{1 - \cos B} \]

### Step 1: Substitute a value for B
To simplify the calculations, we can assign a convenient value to \( B \). Let's choose \( B = 90^\circ \).

### Step 2: Calculate \( \sin B \) and \( \cos B \)
Using \( B = 90^\circ \):
- \( \sin 90^\circ = 1 \)
- \( \cos 90^\circ = 0 \)

### Step 3: Substitute values into the equation
Now substitute these values into the equation:

\[ \cot A = \frac{1}{1 - 0} = 1 \]

### Step 4: Determine the value of A
From \( \cot A = 1 \), we know that:

\[ A = 45^\circ \]

### Step 5: Calculate \( 2A \)
Now, we need to find \( \cot 2A \):

\[ 2A = 2 \times 45^\circ = 90^\circ \]

### Step 6: Find \( \cot 90^\circ \)
We know that:

\[ \cot 90^\circ = 0 \]

### Conclusion
Thus, the value of \( \cot 2A \) is:

\[ \cot 2A = 0 \]

### Final Answer
The value of \( \cot 2A \) is \( 0 \).

---