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

Answer

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

### Step 1: Substitute \( x \) with \( \sec(\theta) \)
Let \( x = \sec(\theta) \). This substitution is valid because for \( x < -1 \), \( \sec(\theta) \) can take negative values when \( \theta \) is in the second quadrant.

### Step 2: Express \( x^2 \)
From the substitution, we have:
\[ x^2 = \sec^2(\theta) \]

### Step 3: Simplify \( \sqrt{x^2 - 1} \)
Using the identity \( \sec^2(\theta) - 1 = \tan^2(\theta) \), we can rewrite:
\[ \sqrt{x^2 - 1} = \sqrt{\sec^2(\theta) - 1} = \sqrt{\tan^2(\theta)} = |\tan(\theta)| \]
Since \( x < -1 \), \( \theta \) is in the second quadrant where \( \tan(\theta) \) is negative. Therefore, we have:
\[ \sqrt{x^2 - 1} = -\tan(\theta) \]

### Step 4: Substitute back into the expression
Now, substituting this back into our expression gives:
\[ \cot^{-1} \left( \frac{1}{-\tan(\theta)} \right) = \cot^{-1} \left( -\cot(\theta) \right) \]

### Step 5: Simplify \( \cot^{-1}(-\cot(\theta)) \)
Recall that \( \cot^{-1}(-\cot(\theta)) = \pi - \cot^{-1}(\cot(\theta)) \), which simplifies to:
\[ \pi - \theta \]

### Step 6: Substitute \( \theta \) back in terms of \( x \)
Since we let \( x = \sec(\theta) \), we have:
\[ \theta = \sec^{-1}(x) \]
Thus, we can rewrite our expression as:
\[ \pi - \sec^{-1}(x) \]

### Final Result
Therefore, the simplified form of the expression is:
\[ \cot^{-1} \left( \frac{1}{\sqrt{x^2 - 1}} \right) = \pi - \sec^{-1}(x) \]

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