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We are asked to prove the following identity:

\[
\frac{2 \sin \beta \cdot \cos \beta}{1 + \cos^2 \beta - \sin^2 \beta} = \tan \beta
\]

### Step-by-Step Proof:

1. **Simplifying the numerator:**

   The numerator is \(2 \sin \beta \cdot \cos \beta\). We can use the trigonometric identity:

   \[
   2 \sin \beta \cdot \cos \beta = \sin(2\beta)
   \]

   So, the expression becomes:

   \[
   \frac{\sin(2\beta)}{1 + \cos^2 \beta - \sin^2 \beta}
   \]

2. **Simplifying the denominator:**

   The denominator is \(1 + \cos^2 \beta - \sin^2 \beta\). Notice that:

   \[
   \cos^2 \beta - \sin^2 \beta = \cos(2\beta)
   \]

   Therefore, the denominator becomes:

   \[
   1 + \cos(2\beta)
   \]

3. **Using a known identity:**

   Recall the double angle identity for cosine:

   \[
   1 + \cos(2\beta) = 2 \cos^2 \beta
   \]

   So, the denominator is \(2 \cos^2 \beta\).

   Now, the expression becomes:

   \[
   \frac{\sin(2\beta)}{2 \cos^2 \beta}
   \]

4. **Simplifying further:**

   We know that:

   \[
   \frac{\sin(2\beta)}{2 \cos^2 \beta} = \frac{2 \sin \beta \cos \beta}{2 \cos^2 \beta} = \frac{\sin \beta}{\cos \beta} = \tan \beta
   \]

Thus, the left-hand side simplifies to \(\tan \beta\), which is equal to the right-hand side, as required.

\[
\boxed{\text{Proven!}}
\]

---

Would you like a more detailed breakdown of any step, or do you have any questions about the proof? Here are 5 follow-up questions related to this topic:

1. How can the double-angle identity for sine be derived from basic trigonometric principles?
2. Why is the double-angle identity for cosine useful in simplifying trigonometric expressions?
3. Can you explain the relationship between the tangent function and the sine and cosine functions?
4. What are the other forms of the double-angle identities for sine and cosine?
5. How would this proof change if we were asked to prove an identity involving cotangent instead of tangent?

**Tip:** Always remember to check for trigonometric identities like double angles or Pythagorean identities when simplifying complex expressions.

Prove that (2 * sin(β) * cos(β)) / (1 + cos^2(β) - sin^2(β)) = tan(β)

This page walks through a step-by-step proof of the trigonometric identity (2 * sin(β) * cos(β)) / (1 + cos^2(β) - sin^2(β)) = tan(β). It covers key concepts in trigonometry, including double angle formulas for sine and cosine, and simplification techniques using trigonometric identities. Suitable for advanced high school or early college students.

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