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Write in terms of of the following double angled trigonometric expression: 

The expression you've provided is:

\[
\frac{\tan 2\theta}{\cos 2\theta} \cdot (1 - \sin 2\theta)
\]

We want to rewrite this expression using double-angle trigonometric identities.

### Step 1: Expressing \(\tan 2\theta\) and \(\cos 2\theta\) in terms of \(\sin \theta\) and \(\cos \theta\)

We know that:
\[
\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}
\]
and
\[
\cos 2\theta = \cos^2 \theta - \sin^2 \theta
\]

However, let's express \(\tan 2\theta\) in terms of \(\sin 2\theta\) and \(\cos 2\theta\):
\[
\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}
\]

### Step 2: Substituting into the given expression

Substituting \(\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}\) into the original expression:

\[
\frac{\frac{\sin 2\theta}{\cos 2\theta}}{\cos 2\theta} \cdot (1 - \sin 2\theta)
\]

This simplifies to:

\[
\frac{\sin 2\theta}{\cos^2 2\theta} \cdot (1 - \sin 2\theta)
\]

### Final Expression
Thus, the expression in terms of \(\sin 2\theta\) and \(\cos 2\theta\) is:

\[
\frac{\sin 2\theta \cdot (1 - \sin 2\theta)}{\cos^2 2\theta}
\]

This is the simplified expression in terms of double angles.

Would you like a detailed explanation on any part of this process, or do you have any questions? Here are some related questions you might find interesting:

1. How do you derive the identity for \(\tan 2\theta\)?
2. What are other forms of double-angle identities?
3. How do you express \(\sin 2\theta\) in terms of \(\sin \theta\) and \(\cos \theta\)?
4. What happens if \(\sin 2\theta = 1\) in the given expression?
5. Can this expression be simplified further under certain conditions?
6. How is \(\tan^2 \theta\) related to \(\sin^2 \theta\) and \(\cos^2 \theta\)?
7. What is the significance of double-angle identities in trigonometry?
8. How does the expression change for \(\theta = 45^\circ\)?

**Tip:** When simplifying trigonometric expressions, it’s often helpful to use well-known identities like double-angle or half-angle formulas to rewrite the expression in a more manageable form.

Learn how to rewrite a trigonometric expression using double-angle identities. Step-by-step explanation with examples. Suitable for advanced high school students and anyone interested in trigonometry.

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