find the value of `tan^(-1){cot(-1/4)}`

Explanation

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

### Step 1: Define the expression
Let \( y = \tan^{-1}(\cot(-\frac{1}{4})) \).

### Step 2: Use the cotangent identity
We know that \( \cot(-\theta) = -\cot(\theta) \). Therefore, we can rewrite:
\[ \cot(-\frac{1}{4}) = -\cot(\frac{1}{4}). \]
Thus, we have:
\[ y = \tan^{-1}(-\cot(\frac{1}{4})). \]

### Step 3: Use the tangent identity
Recall that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Therefore:
\[ -\cot(\frac{1}{4}) = -\frac{1}{\tan(\frac{1}{4})}. \]
This means:
\[ y = \tan^{-1}\left(-\frac{1}{\tan(\frac{1}{4})}\right). \]

### Step 4: Use the tangent of the complementary angle
We know that \( \tan(\frac{\pi}{2} - \theta) = \cot(\theta) \). Hence:
\[ -\cot(\frac{1}{4}) = -\tan\left(\frac{\pi}{2} - \frac{1}{4}\right) = -\tan\left(\frac{2\pi}{4} - \frac{1}{4}\right) = -\tan\left(\frac{1}{4}\right). \]
So we can write:
\[ y = \tan^{-1}\left(-\tan\left(\frac{1}{4}\right)\right). \]

### Step 5: Apply the inverse tangent property
Using the property that \( \tan^{-1}(-x) = -\tan^{-1}(x) \), we have:
\[ y = -\tan^{-1}\left(\tan\left(\frac{1}{4}\right)\right). \]

### Step 6: Simplify using the inverse tangent
Since \( \tan^{-1}(\tan(x)) = x \) for \( x \) in the principal range of \( \tan^{-1} \), we find:
\[ y = -\frac{1}{4}. \]

### Final Result
Thus, the value of \( \tan^{-1}(\cot(-\frac{1}{4})) \) is:
\[ \boxed{-\frac{1}{4}}. \]