https://file.aiquickdraw.com/mathbot/file/17289837966224scz35qo.jpg

Let's solve the given problem step by step.

The problem is asking to find \( \left( \frac{dy}{dx} \right)^2 \) given that \( x = \sin(y) \).

### Step 1: Differentiate \( x = \sin(y) \)
We need to implicitly differentiate both sides with respect to \( x \).

\[
\frac{d}{dx} \left( x \right) = \frac{d}{dx} \left( \sin(y) \right)
\]

This gives:

\[
1 = \cos(y) \cdot \frac{dy}{dx}
\]

### Step 2: Solve for \( \frac{dy}{dx} \)
From the equation \( 1 = \cos(y) \cdot \frac{dy}{dx} \), solve for \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} = \frac{1}{\cos(y)}
\]

### Step 3: Find \( \left( \frac{dy}{dx} \right)^2 \)
Now, we square both sides to find \( \left( \frac{dy}{dx} \right)^2 \):

\[
\left( \frac{dy}{dx} \right)^2 = \frac{1}{\cos^2(y)}
\]

### Step 4: Express \( \cos^2(y) \) in terms of \( x \)
From the given condition \( x = \sin(y) \), we know that:

\[
\cos^2(y) = 1 - \sin^2(y) = 1 - x^2
\]

Thus,

\[
\left( \frac{dy}{dx} \right)^2 = \frac{1}{1 - x^2}
\]

### Conclusion:
The correct answer is option (a) \( \frac{1}{1 - x^2} \).

Let me know if you need further clarification or have any questions. 

Here are five related questions to expand your understanding:
1. How do you differentiate implicitly when the equation involves trigonometric functions?
2. What are the main steps to finding derivatives of inverse trigonometric functions?
3. How do you apply the chain rule in differentiation?
4. Why is \( \cos^2(y) \) written as \( 1 - \sin^2(y) \) using trigonometric identities?
5. Can you think of a real-world application of this type of trigonometric differentiation?

**Tip:** When solving implicit differentiation problems, always apply the chain rule carefully when differentiating functions of other variables!

Learn how to solve the problem involving implicit differentiation when x = sin(y). This solution covers the chain rule and the use of trigonometric identities, leading to (dy/dx)^2 = 1/(1 - x^2). Ideal for undergraduate calculus students.

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