The expression you have written:

\[
\frac{\tan \frac{c}{2}}{1 + \tan^2 \frac{c}{2}}
\]

can be simplified using a known trigonometric identity. This is actually a form of the identity for \(\sin c\), which is derived from the half-angle formulae. Here's the detailed process:

### Simplification:

We know the following identity for sine in terms of tangent half-angle:

\[
\sin c = \frac{2 \tan \frac{c}{2}}{1 + \tan^2 \frac{c}{2}}
\]

The expression \(\frac{\tan \frac{c}{2}}{1 + \tan^2 \frac{c}{2}}\) simplifies to:

\[
\frac{1}{2} \sin c
\]

Thus, the simplified form of your expression is:

\[
\frac{1}{2} \sin c
\]

### Conclusion:
The given expression \(\frac{\tan \frac{c}{2}}{1 + \tan^2 \frac{c}{2}}\) simplifies to \(\frac{1}{2} \sin c\).

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Would you like further details on this simplification or have any questions?

Here are 5 related questions:
1. What is the general form of the half-angle identity for sine and cosine?
2. How do you derive the half-angle formula from basic trigonometric identities?
3. What other trigonometric identities can be simplified using half-angle formulas?
4. Can the half-angle identities be applied to solve integral problems in calculus?
5. How can you express \(\cos c\) using \(\tan \frac{c}{2}\)?

**Tip:** Understanding half-angle identities can help simplify complex trigonometric problems and make integration easier in calculus.

Simplify the trigonometric expression \(\frac{\tan \frac{c}{2}}{1 + \tan^2 \frac{c}{2}}\) using the half-angle identity. Learn how this expression simplifies to \(\frac{1}{2} \sin c\), with a detailed breakdown of the half-angle formula for sine. Ideal for high school students in Grades 10-12.

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