The principle value of `"tan"^(-1)("cot"(43pi)/(4))` is

Options

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

Answer

To find the principal value of \( \tan^{-1} \left( \cot \left( \frac{43\pi}{4} \right) \right) \), we can follow these steps:

### Step 1: Simplify the Angle
First, we need to simplify the angle \( \frac{43\pi}{4} \). We can express this angle in terms of \( 2\pi \) since the cotangent function has a period of \( \pi \).

\[ \frac{43\pi}{4} = 10\pi + \frac{3\pi}{4} \]

Since \( \cot(x) \) has a period of \( \pi \), we can reduce \( \frac{43\pi}{4} \) by subtracting \( 10\pi \):

\[ \cot \left( \frac{43\pi}{4} \right) = \cot \left( \frac{3\pi}{4} \right) \]

### Step 2: Evaluate \( \cot \left( \frac{3\pi}{4} \right) \)
Next, we evaluate \( \cot \left( \frac{3\pi}{4} \right) \). The cotangent function is defined as:

\[ \cot(x) = \frac{1}{\tan(x)} \]

We know that:

\[ \cot \left( \frac{3\pi}{4} \right) = -1 \]

### Step 3: Substitute into the Inverse Tangent Function
Now we substitute \( -1 \) back into the inverse tangent function:

\[ \tan^{-1} \left( \cot \left( \frac{43\pi}{4} \right) \right) = \tan^{-1}(-1) \]

### Step 4: Find the Principal Value of \( \tan^{-1}(-1) \)
The principal value of \( \tan^{-1}(-1) \) is:

\[ \tan^{-1}(-1) = -\frac{\pi}{4} \]

### Conclusion
Thus, the principal value of \( \tan^{-1} \left( \cot \left( \frac{43\pi}{4} \right) \right) \) is:

\[ \boxed{-\frac{\pi}{4}} \]