`cot^(-1)(cot 4)=`

Options

A. 4
B. `1/4`
C. `4-pi`
D. `pi-4`

Answer

To solve the problem \( \cot^{-1}(\cot 4) \), we can follow these steps:

### Step 1: Understand the range of the cotangent function
The cotangent function, \( \cot x \), is periodic with a period of \( \pi \). The inverse cotangent function, \( \cot^{-1}(x) \), returns values in the range \( (0, \pi) \).

### Step 2: Determine the equivalent angle
Since \( 4 \) is not in the range \( (0, \pi) \), we need to find an equivalent angle that is within this range. We can express \( 4 \) in terms of \( \pi \):
- \( 4 \) is approximately \( 1.273 \pi \) (since \( \pi \approx 3.14 \)).
- To find an equivalent angle in the range \( (0, \pi) \), we can subtract \( \pi \) from \( 4 \):
\[ 4 - \pi \approx 4 - 3.14 = 0.86 \]

### Step 3: Apply the cotangent identity
Using the property of the cotangent function:
\[ \cot^{-1}(\cot x) = x \quad \text{if } x \in (0, \pi) \]
we can conclude that:
\[ \cot^{-1}(\cot 4) = 4 - \pi \]

### Step 4: Final answer
Thus, the final answer is:
\[ \cot^{-1}(\cot 4) = 4 - \pi \]