how is sin^2 times x / x equal to sin x over x times x

Let's analyze this step by step.

You have the expression:
\[
\frac{\sin^2(x)}{x}
\]
and you are asking how it's related to:
\[
\frac{\sin(x)}{x} \times x
\]

### Step 1: Understanding the expressions

- The first expression is \(\frac{\sin^2(x)}{x}\). This means \(\frac{(\sin(x))^2}{x}\), or equivalently, \(\sin(x) \cdot \frac{\sin(x)}{x}\).

- The second expression is \(\frac{\sin(x)}{x} \times x\), which simplifies to just \(\sin(x)\), since \(\frac{\sin(x)}{x} \times x = \sin(x)\).

### Step 2: Comparing the two expressions

1. The first expression \(\frac{\sin^2(x)}{x}\) is the same as:
   \[
   \frac{(\sin(x))^2}{x} = \sin(x) \cdot \frac{\sin(x)}{x}
   \]
   
2. The second expression \(\frac{\sin(x)}{x} \times x\) simplifies to:
   \[
   \sin(x)
   \]

### Step 3: Are they equal?

The first expression is \(\sin(x) \cdot \frac{\sin(x)}{x}\), while the second simplifies directly to \(\sin(x)\). For them to be equal, you would need to simplify the first expression further.

So, \(\sin(x) \cdot \frac{\sin(x)}{x} = \sin(x)\), meaning the two expressions are **not generally equal**, but the two expressions may simplify similarly depending on context (for example, as \(x\) approaches 0, \(\frac{\sin(x)}{x}\) approaches 1, making the comparison clearer for certain limits).

### Conclusion

The two expressions are not always equal, but under certain conditions, such as limits, their behavior can be comparable.

Would you like more details on where these expressions might appear in limits or other contexts? Here are some related questions:

1. What is the behavior of \(\frac{\sin(x)}{x}\) as \(x \to 0\)?
2. How does L'Hopital's Rule help in evaluating limits involving \(\sin(x)\)?
3. How can you prove that \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\)?
4. What is the significance of trigonometric limits in calculus?
5. How does Taylor series expansion relate to simplifying trigonometric expressions?

**Tip:** Whenever you see trigonometric expressions like \(\frac{\sin(x)}{x}\), limits and small-angle approximations are often useful tools to analyze their behavior.

Explore the mathematical relationship between sin^2(x)/x and sin(x)/x times x, including trigonometric and limit concepts. This solution discusses key formulas and theorems such as L'Hopital's Rule and the trigonometric limit theorem, suitable for students in Grades 11-12 and introductory calculus courses.

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