`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: Let \( \theta = \cot^{-1} x \)
This means that \( \cot \theta = x \).

### Step 2: Use the definition of cotangent
From the definition of cotangent, we know that:
\[ \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{x}{1} \]
This means 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 \( h \):
\[ h = \sqrt{(\text{adjacent})^2 + (\text{opposite})^2} = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \]

### Step 4: Calculate \( \sin \theta \)
Now, we can find \( \sin \theta \) using the definition of sine:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{x^2 + 1}} \]

### Step 5: Substitute back to find \( \sin(\cot^{-1} x) \)
Since we defined \( \theta = \cot^{-1} x \), we have:
\[ \sin(\cot^{-1} x) = \sin \theta = \frac{1}{\sqrt{x^2 + 1}} \]

### Final Result
Thus, the solution to the problem is:
\[ \sin(\cot^{-1} x) = \frac{1}{\sqrt{x^2 + 1}} \]

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