`d/(dx)(tan^(-1) ((cos x)/(1+sinx))`=

Options

A. `-1/2`
B. `1/2`
C. `-1`
D. 1

Answer

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

### Step 1: Rewrite the function
Let:
\[ y = \tan^{-1}\left(\frac{\cos x}{1 + \sin x}\right) \]

### Step 2: Use trigonometric identities
We can use the identity that relates sine and cosine:
\[ \frac{\cos x}{1 + \sin x} = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) \]
This is derived from the double angle formulas and properties of sine and cosine.

### Step 3: Substitute the identity into the function
Thus, we can rewrite \( y \):
\[ y = \tan^{-1}\left(\tan\left(\frac{\pi}{4} - \frac{x}{2}\right)\right) \]
Since the tangent and arctangent functions are inverses, we have:
\[ y = \frac{\pi}{4} - \frac{x}{2} \]

### Step 4: Differentiate with respect to \( x \)
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 0 - \frac{1}{2} = -\frac{1}{2} \]

### Final Answer
Thus, the derivative is:
\[ \frac{dy}{dx} = -\frac{1}{2} \]