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Let's solve the problem step by step.

We are given:
- \( f(x) = 1 - x^2 \)
- \( g(x) = \sin(x) \)

The question asks us to find the expression for \(\cos^2(g(x))\).

### Step 1: Substitute \( g(x) \) into the expression
Since \( g(x) = \sin(x) \), we have:
\[
g(x) = \sin(x)
\]

Now, substitute \( g(x) \) into \(\cos^2(g(x))\):
\[
\cos^2(g(x)) = \cos^2(\sin(x))
\]

### Step 2: Use the identity provided
The identity given is:
\[
\sin^2(z) + \cos^2(z) = 1
\]
This identity holds for any \( z \). Here, our \( z \) is \( \sin(x) \).

### Step 3: Determine the correct option
Since \(\cos^2(g(x))\) translates to \(\cos^2(\sin(x))\), we can compare the expressions given in the options:

Given the possible answers:
1. \( f(x)g(x) \)
2. \( \frac{f(x)}{g(x)}, g(x) \neq 0 \)
3. \( f(g(x)) \)
4. \( (g \circ g)(x) \)
5. \( (g \circ f)(x) \)

Let's substitute \( f(x) = 1 - x^2 \) and \( g(x) = \sin(x) \) in each option:

- \( f(g(x)) = f(\sin(x)) = 1 - \sin^2(x) \)
- Using the identity \(\sin^2(x) + \cos^2(x) = 1\), we get:
  \[
  f(g(x)) = \cos^2(x)
  \]

### Final Conclusion
The correct option is \( f(g(x)) \), which matches with \(\cos^2(\sin(x))\).

So, the correct answer is **Option 3: \( f(g(x)) \)**.

Would you like more details on any part of this problem? Here are some follow-up questions:

1. How does substituting a function inside another function work?
2. Can you explain the trigonometric identity \( \sin^2(z) + \cos^2(z) = 1 \) in more detail?
3. How do you identify the correct option in a multiple-choice question involving functions?
4. What is the importance of domain restrictions like \( g(x) \neq 0 \)?
5. Can \( g(x) \) be any other function, or is it restricted to trigonometric functions?

**Tip:** When dealing with trigonometric identities, always keep the fundamental identities in mind, as they often simplify complex-looking expressions.

If \( f(x) = 1 - x^2 \) and \( g(x) = \sin(x) \), then \( \cos^2(g(x)) \) is?

This problem explores the composition of functions and the application of trigonometric identities to find the expression for \( \cos^2(g(x)) \). It is designed for high school students and covers important concepts in both algebra and trigonometry.

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