`sin(cot^(-1)x)=`

Options

A. `sqrt((1+x^2)`
B. x
C. `(1+x^2)^(-3//2)`
D. `(1+x^2)^(-1//2)`

Answer

To solve the problem \( \sin(\cot^{-1} x) \), we can follow these steps:

### Step 1: Understand the Inverse Function
The expression \( \cot^{-1} x \) represents an angle \( \theta \) such that \( \cot \theta = x \). This means that:
\[ \theta = \cot^{-1} x \]

### Step 2: Use the Cotangent Identity
From the definition of cotangent, we have:
\[ \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = x \]
This implies that we can represent the sides of a right triangle where the adjacent side is \( x \) and the opposite side is \( 1 \).

### Step 3: Find the Hypotenuse
Using the Pythagorean theorem, we can find the hypotenuse \( h \):
\[ h = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \]

### Step 4: Find the Sine of the Angle
Now, we want to find \( \sin \theta \). Recall that:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{x^2 + 1}} \]

### Step 5: Write the Final Answer
Thus, we have:
\[ \sin(\cot^{-1} x) = \frac{1}{\sqrt{x^2 + 1}} \]

### Summary of Steps
1. Recognize \( \theta = \cot^{-1} x \).
2. Set up a right triangle with adjacent side \( x \) and opposite side \( 1 \).
3. Calculate the hypotenuse using the Pythagorean theorem.
4. Use the definition of sine to find \( \sin \theta \).
5. Write the final expression.