Simplify
`cot^(-1)((1)/(sqrt(x^(2)-1))), x lt -1 `

Answer

To simplify the expression \( \cot^{-1}\left(\frac{1}{\sqrt{x^2 - 1}}\right) \) given that \( x < -1 \), we can follow these steps:

### Step 1: Rewrite \( x \) in terms of \( \sec \theta \)
Since \( x < -1 \), we can let \( x = \sec \theta \) where \( \theta \) is in the range \( \left(\frac{\pi}{2}, \pi\right) \).

### Step 2: Substitute \( x \) into the expression
Substituting \( x = \sec \theta \) into the expression gives:
\[ \cot^{-1}\left(\frac{1}{\sqrt{\sec^2 \theta - 1}}\right) \]

### Step 3: Simplify the expression under the square root
Using the identity \( \sec^2 \theta - 1 = \tan^2 \theta \), we can rewrite the expression as:
\[ \cot^{-1}\left(\frac{1}{\sqrt{\tan^2 \theta}}\right) \]

### Step 4: Simplify further
Since \( \sqrt{\tan^2 \theta} = |\tan \theta| \) and \( \tan \theta \) is negative in the interval \( \left(\frac{\pi}{2}, \pi\right) \), we have:
\[ \sqrt{\tan^2 \theta} = -\tan \theta \]
Thus, the expression becomes:
\[ \cot^{-1}\left(\frac{1}{-\tan \theta}\right) = \cot^{-1}\left(-\cot \theta\right) \]

### Step 5: Use the property of inverse cotangent
Using the property \( \cot^{-1}(-x) = \pi - \cot^{-1}(x) \), we can rewrite the expression as:
\[ \pi - \cot^{-1}(\cot \theta) \]

### Step 6: Simplify \( \cot^{-1}(\cot \theta) \)
Since \( \cot^{-1}(\cot \theta) = \theta \) (for \( \theta \) in the appropriate range), we have:
\[ \pi - \theta \]

### Step 7: Substitute back for \( \theta \)
Recall that \( \theta = \sec^{-1}(x) \), so we substitute back to get:
\[ \pi - \sec^{-1}(x) \]

### Final Answer
Thus, the simplified expression is:
\[ \pi - \sec^{-1}(x) \]