If f(x) = 2x/1+x² prove that f (tan theta)= sin2 theta

To prove that if \( f(x) = \frac{2x}{1 + x^2} \), then \( f(\tan \theta) = \sin 2\theta \), let's proceed step by step.

### Step 1: Express \( f(\tan \theta) \)
We know:
\[
f(x) = \frac{2x}{1 + x^2}
\]
Substitute \( x = \tan \theta \) into the function:
\[
f(\tan \theta) = \frac{2 \tan \theta}{1 + \tan^2 \theta}
\]

### Step 2: Use trigonometric identities
Now use the trigonometric identity:
\[
1 + \tan^2 \theta = \sec^2 \theta
\]
Thus, the expression becomes:
\[
f(\tan \theta) = \frac{2 \tan \theta}{\sec^2 \theta}
\]

### Step 3: Simplify using basic trigonometric formulas
Recall that \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \), so:
\[
f(\tan \theta) = 2 \tan \theta \cdot \cos^2 \theta
\]

Now use the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), so:
\[
f(\tan \theta) = 2 \cdot \frac{\sin \theta}{\cos \theta} \cdot \cos^2 \theta
\]

Simplifying:
\[
f(\tan \theta) = 2 \sin \theta \cdot \cos \theta
\]

### Step 4: Apply the double-angle identity for sine
We know the double-angle identity for sine:
\[
\sin 2\theta = 2 \sin \theta \cdot \cos \theta
\]

Thus, we can conclude:
\[
f(\tan \theta) = \sin 2\theta
\]

### Conclusion
We have shown that:
\[
f(\tan \theta) = \sin 2\theta
\]
as required.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. What are the double-angle identities for sine and cosine?
2. How do you derive trigonometric identities using triangles?
3. What is the relationship between \( \tan \theta \) and \( \sin \theta \), \( \cos \theta \)?
4. How do the addition formulas for trigonometric functions work?
5. Can you express higher powers of trigonometric functions using identities?

**Tip:** When working with trigonometric functions and identities, always check if common identities like double-angle or Pythagorean identities can simplify the expression.

Learn how to prove the function f(tan theta) equals sin2 theta using trigonometric identities. This problem involves applying the double-angle identity for sine and basic trigonometric formulas. Suitable for advanced high school students or college-level trigonometry courses.

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