Express `cot(cosec^(-1)x)` as an algebraic function of `x`.

Explanation

To express \( \cot(\csc^{-1} x) \) as an algebraic function of \( x \), we can follow these steps:

### Step 1: Define the angle
Let \( \theta = \csc^{-1}(x) \). By the definition of the cosecant function, we have:
\[ x = \csc(\theta) = \frac{1}{\sin(\theta)} \]

### Step 2: Find the sine
From the equation \( x = \csc(\theta) \), we can express \( \sin(\theta) \):
\[ \sin(\theta) = \frac{1}{x} \]

### Step 3: Use the Pythagorean identity
Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), we can find \( \cos(\theta) \):
\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(\frac{1}{x}\right)^2 = 1 - \frac{1}{x^2} = \frac{x^2 - 1}{x^2} \]
Taking the square root gives us:
\[ \cos(\theta) = \sqrt{\frac{x^2 - 1}{x^2}} = \frac{\sqrt{x^2 - 1}}{|x|} \]

### Step 4: Find cotangent
Now we can find \( \cot(\theta) \):
\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\frac{\sqrt{x^2 - 1}}{|x|}}{\frac{1}{x}} = \frac{\sqrt{x^2 - 1}}{|x|} \cdot x = \frac{\sqrt{x^2 - 1}}{|x|} \]

### Step 5: Determine the sign
We need to consider the sign of \( x \):
- If \( x \geq 1 \), then \( |x| = x \) and \( \cot(\csc^{-1}(x)) = \frac{\sqrt{x^2 - 1}}{x} \).
- If \( x \leq -1 \), then \( |x| = -x \) and \( \cot(\csc^{-1}(x)) = \frac{\sqrt{x^2 - 1}}{-x} = -\frac{\sqrt{x^2 - 1}}{x} \).

### Final Result
Thus, we can summarize the result as:
\[ \cot(\csc^{-1}(x)) = \begin{cases} \frac{\sqrt{x^2 - 1}}{x} & \text{if } x \geq 1 \\ -\frac{\sqrt{x^2 - 1}}{x} & \text{if } x \leq -1 \end{cases} \]