` y = (1 -tan^(2) (x//2))/(1 + tan^(2)(x//2)) , " find " (dy)/(dx)`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function

\[ y = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}, \]

we can use a trigonometric identity.

### Step 1: Recognize the Trigonometric Identity
The expression resembles the double angle formula for cosine, which states:

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

In our case, we can let \( \theta = \frac{x}{2} \). Therefore, we can rewrite \( y \) as:

\[ y = \cos(x). \]

### Step 2: Differentiate the Function
Now, we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = \frac{d}{dx}(\cos(x)). \]

Using the derivative of cosine, we have:

\[ \frac{dy}{dx} = -\sin(x). \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = -\sin(x). \]

---