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The given expression is:

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

We are tasked with rewriting it in terms of \(\tan(\theta)\). Let's proceed step by step using the double-angle identities for sine, cosine, and tangent.

### Step 1: Double-Angle Identities
1. \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\)
2. \(\cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}\)
3. \(\sin(2\theta) = \frac{2\tan(\theta)}{1 + \tan^2(\theta)}\)

### Step 2: Substitute into the Expression

We substitute these identities into the original expression:

\[
\frac{\frac{2\tan(\theta)}{1 - \tan^2(\theta)}}{\frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}\left(1 - \frac{2\tan(\theta)}{1 + \tan^2(\theta)}\right)}
\]

### Step 3: Simplify the Denominator

Now, simplify the term inside the parentheses:

\[
1 - \frac{2\tan(\theta)}{1 + \tan^2(\theta)} = \frac{1 + \tan^2(\theta) - 2\tan(\theta)}{1 + \tan^2(\theta)}
\]

Thus, the entire denominator becomes:

\[
\frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)} \cdot \frac{1 + \tan^2(\theta) - 2\tan(\theta)}{1 + \tan^2(\theta)} = \frac{(1 - \tan^2(\theta))(1 + \tan^2(\theta) - 2\tan(\theta))}{(1 + \tan^2(\theta))^2}
\]

### Step 4: Final Simplification

The numerator is still:

\[
\frac{2\tan(\theta)}{1 - \tan^2(\theta)}
\]

So the entire expression becomes:

\[
\frac{\frac{2\tan(\theta)}{1 - \tan^2(\theta)}}{\frac{(1 - \tan^2(\theta))(1 + \tan^2(\theta) - 2\tan(\theta))}{(1 + \tan^2(\theta))^2}}
\]

Canceling \((1 - \tan^2(\theta))\) from both numerator and denominator, the expression simplifies to:

\[
\frac{2\tan(\theta)(1 + \tan^2(\theta))^2}{(1 + \tan^2(\theta) - 2\tan(\theta))}
\]

This is the expression rewritten in terms of \(\tan(\theta)\).

Would you like further clarifications or details?

Here are 5 related questions to expand understanding:

1. What is the significance of double-angle identities in trigonometry?
2. How does simplifying complex trigonometric expressions help in solving real-world problems?
3. Can you derive the double-angle identity for sine using basic trigonometric identities?
4. What are some common applications of the tangent function in mathematics and physics?
5. How do trigonometric identities aid in simplifying integral and differential equations?

**Tip**: When dealing with double-angle identities, always ensure you correctly apply the corresponding formula for sine, cosine, or tangent.

Write in terms of tanθ the following double-angled trigonometric expression: \(\frac{\tan(2\theta)}{\cos(2\theta)(1 - \sin(2\theta))}\)

Learn how to express the trigonometric double-angled expression \(\frac{\tan(2\theta)}{\cos(2\theta)(1 - \sin(2\theta))}\) in terms of \(\tan(\theta)\). This detailed step-by-step solution utilizes double-angle identities for sine, cosine, and tangent. Suitable for students in Grades 10-12.

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