y=`tan^(-1)((sinx)/(1+cosx))` Find dy/dx

Explanation

To find the derivative \( \frac{dy}{dx} \) for the function \( y = \tan^{-1}\left(\frac{\sin x}{1 + \cos x}\right) \), we can follow these steps:

### Step 1: Simplify the expression inside the arctangent
We can use the trigonometric identities to simplify the expression. Recall that:
- \( \sin x = 2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) \)
- \( 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \)

Substituting these into the expression gives:
...