Let's simplify the given expression step by step.

The expression is:

\[
(1 - \sin 2x)(1 + \tan^2 x)
\]

### Step 1: Use a trigonometric identity for \(\tan^2 x\).
We know that:

\[
1 + \tan^2 x = \sec^2 x
\]

Thus, the expression becomes:

\[
(1 - \sin 2x) \cdot \sec^2 x
\]

### Step 2: Leave it as is for now.
At this point, there isn't a more straightforward simplification unless you substitute \(\sin 2x\) using the double-angle identity (\(\sin 2x = 2 \sin x \cos x\)). However, the expression \((1 - \sin 2x) \cdot \sec^2 x\) is already simplified in terms of basic trigonometric functions.

Would you like to further simplify or substitute \(\sin 2x\) with its identity?

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Let me know if you need more details or have any questions!

Here are 5 related questions:
1. How do you derive the identity \(1 + \tan^2 x = \sec^2 x\)?
2. What is the double-angle identity for \(\sin 2x\)?
3. How does the expression change if you substitute \(\sin 2x\) with \(2 \sin x \cos x\)?
4. What happens if you expand the expression instead of simplifying it?
5. Can this expression be integrated over a certain interval?

**Tip**: Knowing the fundamental trigonometric identities can help you simplify complex expressions more easily.

Learn how to simplify the trigonometric expression (1 - sin 2x)(1 + tan^2 x) using fundamental trigonometric identities. This problem involves the Pythagorean identity and the double-angle identity for sin 2x. Suitable for students in Grades 10-12.

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