What is `(1-"tan"^(2)(x)/(2))/(1+"tan"^(2)(x)/(2))`equal to :

Options

A. sinx . cosx
B. tanx
C. sin x
D. cos x

Explanation

To solve the expression \(\frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}\), we can use a known trigonometric identity.

### Step-by-step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \]

2. **Use the Trigonometric Identity**:
We recognize that the expression \(\frac{1 - \tan^2\theta}{1 + \tan^2\theta}\) is a standard trigonometric identity that equals \(\cos(2\theta)\). Here, we let \(\theta = \frac{x}{2}\).

3. **Apply the Identity**:
Therefore, we can rewrite our expression as:
\[ \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} = \cos\left(2 \cdot \frac{x}{2}\right) = \cos(x) \]

4. **Final Result**:
Thus, we conclude that:
\[ \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} = \cos(x) \]

### Conclusion:
The final answer is:
\[ \cos(x) \]