The principal value of `cot^(-1)(-1)` is

Options

A. `-pi/4`
B. `pi/4`
C. `(5pi)/4`
D. `(3pi)/4`

Explanation

To find the principal value of \( \cot^{-1}(-1) \), we can follow these steps:

### Step 1: Set up the equation
Let \( x = \cot^{-1}(-1) \). By definition, this means that:
\[ \cot(x) = -1 \]
where \( x \) is in the interval \( (0, \pi) \).

### Step 2: Determine the angle
We know that:
\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]
For \( \cot(x) = -1 \), it implies that:
\[ \cos(x) = -\sin(x) \]
This occurs when \( x \) is in the second quadrant, where the cosine is negative and sine is positive.

### Step 3: Find the reference angle
The reference angle where \( \cot(x) = 1 \) is \( \frac{\pi}{4} \). Therefore, in the second quadrant, the angle \( x \) can be expressed as:
\[ x = \pi - \frac{\pi}{4} \]

### Step 4: Calculate the angle
Calculating this gives:
\[ x = \pi - \frac{\pi}{4} = \frac{4\pi}{4} - \frac{\pi}{4} = \frac{3\pi}{4} \]

### Step 5: Conclusion
Thus, the principal value of \( \cot^{-1}(-1) \) is:
\[ \boxed{\frac{3\pi}{4}} \]

---