Evaluate the following
`tan^(-1) { cot (- 1/4)}`

Answer

To evaluate \( \tan^{-1} \left( \cot \left( -\frac{1}{4} \right) \right) \), we can follow these steps:

### Step 1: Use the property of cotangent
We know that:
\[ \cot(-\theta) = -\cot(\theta) \]
Thus, we can rewrite:
\[ \cot\left(-\frac{1}{4}\right) = -\cot\left(\frac{1}{4}\right) \]

### Step 2: Substitute into the inverse tangent function
Now we substitute this into our original expression:
\[ \tan^{-1} \left( \cot\left(-\frac{1}{4}\right) \right) = \tan^{-1} \left( -\cot\left(\frac{1}{4}\right) \right) \]

### Step 3: Use the property of inverse tangent
We know that:
\[ \tan^{-1}(-x) = -\tan^{-1}(x) \]
Applying this property, we get:
\[ \tan^{-1} \left( -\cot\left(\frac{1}{4}\right) \right) = -\tan^{-1} \left( \cot\left(\frac{1}{4}\right) \right) \]

### Step 4: Relate cotangent to tangent
Recall that:
\[ \cot(\theta) = \tan\left(\frac{\pi}{2} - \theta\right) \]
Thus, we can write:
\[ \tan^{-1} \left( \cot\left(\frac{1}{4}\right) \right) = \tan^{-1} \left( \tan\left(\frac{\pi}{2} - \frac{1}{4}\right) \right) \]

### Step 5: Simplify using the inverse tangent property
Since \( \tan^{-1}(\tan(x)) = x \) for \( x \) in the principal range of \( \tan^{-1} \):
\[ \tan^{-1} \left( \tan\left(\frac{\pi}{2} - \frac{1}{4}\right) \right) = \frac{\pi}{2} - \frac{1}{4} \]

### Step 6: Substitute back into the expression
Now substituting this back, we have:
\[ -\tan^{-1} \left( \cot\left(\frac{1}{4}\right) \right) = -\left(\frac{\pi}{2} - \frac{1}{4}\right) \]
This simplifies to:
\[ -\frac{\pi}{2} + \frac{1}{4} \]

### Final Result
Thus, the value of \( \tan^{-1} \left( \cot \left( -\frac{1}{4} \right) \right) \) is:
\[ \frac{1}{4} - \frac{\pi}{2} \]