https://file.aiquickdraw.com/mathbot/file/17267383860643zzy3jq6.jpg

The expression shown in the image is:

\[
\sin(\cot^{-1} x)
\]

This represents the sine of the inverse cotangent (or arccotangent) of \(x\). Let's simplify it.

### Step-by-Step Solution:

1. Let \(\theta = \cot^{-1} x\). This means:
   \[
   \cot(\theta) = x
   \]
   From the definition of cotangent:
   \[
   \cot(\theta) = \frac{1}{\tan(\theta)} = x \quad \text{or} \quad \tan(\theta) = \frac{1}{x}.
   \]
   
2. We need to find \(\sin(\theta)\). We can use a right triangle to represent this. If \(\tan(\theta) = \frac{1}{x}\), then in the corresponding right triangle:
   - The opposite side is 1.
   - The adjacent side is \(x\).
   - The hypotenuse (by the Pythagorean theorem) is:
     \[
     \text{Hypotenuse} = \sqrt{1^2 + x^2} = \sqrt{x^2 + 1}.
     \]
   
3. Now, using the definition of sine:
   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{x^2 + 1}}.
   \]

Thus, the simplified result of \(\sin(\cot^{-1} x)\) is:

\[
\sin(\cot^{-1} x) = \frac{1}{\sqrt{x^2 + 1}}.
\]

Would you like more details on the derivation or have any other questions?

Here are 5 related questions to consider:
1. How do you derive the expression for \(\cos(\cot^{-1} x)\)?
2. What is the relationship between the inverse cotangent and inverse tangent functions?
3. How do you compute the derivative of \(\sin(\cot^{-1} x)\)?
4. How do you express \(\cot^{-1} x\) in terms of \(\arctan(x)\)?
5. What are the domain and range of \(\sin(\cot^{-1} x)\)?

**Tip:** For solving inverse trigonometric functions, using right triangles or their unit circle equivalents often simplifies the work.

What is the simplified form of sin(cot^(-1)(x))?

Learn how to simplify the expression sin(cot^(-1)(x)) using trigonometric relationships and right triangles. This detailed solution is suitable for students in Grades 10-12 and covers key concepts including inverse trigonometric functions and the Pythagorean theorem.

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