In any triangle `A B C ,\ sin A-cos B=cos C ,\ ` the angle`\ B` is a.`pi/2` b. `pi/3` c.`pi/4` d. `pi/6`

Options

A. `pi/2`
B. `pi/3`
C. `pi/4`
D. `pi/6`

Explanation

To solve the problem, we start with the equation given in the triangle \( ABC \):

\[ \sin A - \cos B = \cos C \]

### Step 1: Rearranging the equation
We can rearrange the equation to isolate \( \sin A \):

\[ \sin A = \cos B + \cos C \]

### Step 2: Using the triangle angle sum property
In any triangle, the sum of angles is \( \pi \):

\[ A + B + C = \pi \]

From this, we can express \( C \) in terms of \( A \) and \( B \):

\[ C = \pi - A - B \]

### Step 3: Substitute \( C \) into the equation
Now, we substitute \( C \) into the equation \( \sin A = \cos B + \cos C \):

\[ \sin A = \cos B + \cos(\pi - A - B) \]

Using the property of cosine, \( \cos(\pi - x) = -\cos x \):

\[ \sin A = \cos B - \cos(A + B) \]

### Step 4: Using the cosine addition formula
We can use the cosine addition formula:

\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]

Substituting this into our equation gives:

\[ \sin A = \cos B - (\cos A \cos B - \sin A \sin B) \]

### Step 5: Rearranging the equation
Rearranging gives us:

\[ \sin A + \cos A \cos B - \sin A \sin B = \cos B \]

### Step 6: Factor out \( \sin A \)
Now, we can factor out \( \sin A \):

\[ \sin A (1 + \sin B) = \cos B (1 - \cos A) \]

### Step 7: Analyzing the angles
Since \( A + B + C = \pi \), we can analyze the values of \( B \). We need to check the options given:

1. **Option a: \( \frac{\pi}{2} \)**
2. **Option b: \( \frac{\pi}{3} \)**
3. **Option c: \( \frac{\pi}{4} \)**
4. **Option d: \( \frac{\pi}{6} \)**

### Step 8: Testing \( B = \frac{\pi}{2} \)
If we set \( B = \frac{\pi}{2} \):

- Then \( C = \pi - A - \frac{\pi}{2} = \frac{\pi}{2} - A \).
- Thus, \( \cos B = 0 \) and \( \cos C = \sin A \).

Substituting back into the original equation:

\[ \sin A - 0 = \sin A \]

This holds true. Therefore, \( B = \frac{\pi}{2} \) is a valid solution.

### Conclusion
Thus, the angle \( B \) is:

\[ \boxed{\frac{\pi}{2}} \]