If ` y = tan^(-1)((cos x)/(1 + sin x)), "then" (dy)/(dx) ` is equal to

Options

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

Explanation

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

### Step 1: Simplify the Argument of the Inverse Tangent Function
First, simplify the expression inside the \(\tan^{-1}\) function:
\[ \frac{\cos x}{1 + \sin x} \]

Recall the trigonometric identities:
\[ \cos x = \cos^2\left(\frac{x}{2}\right) - \sin^2\left(\frac{x}{2}\right) \]
\[ 1 + \sin x = \cos^2\left(\frac{x}{2}\right) + \sin^2\left(\frac{x}{2}\right) + 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) \]

### Step 2: Use Trigonometric Identities
Rewrite the numerator and the denominator using half-angle identities:
\[ \cos x = \cos^2\left(\frac{x}{2}\right) - \sin^2\left(\frac{x}{2}\right) \]
\[ 1 + \sin x = \left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)^2 \]

Thus,
\[ \frac{\cos x}{1 + \sin x} = \frac{\cos^2\left(\frac{x}{2}\right) - \sin^2\left(\frac{x}{2}\right)}{\left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)^2} \]

### Step 3: Simplify the Fraction
Factor the numerator using the difference of squares:
\[ \frac{\cos^2\left(\frac{x}{2}\right) - \sin^2\left(\frac{x}{2}\right)}{\left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)^2} = \frac{\left(\cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right)\right)\left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)}{\left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)^2} \]

Cancel out the common term:
\[ \frac{\left(\cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right)\right)\left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)}{\left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)^2} = \frac{\cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)} \]

### Step 4: Express in Terms of Tangent
Divide the numerator and the denominator by \(\cos\left(\frac{x}{2}\right)\):
\[ \frac{\cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)} = \frac{1 - \tan\left(\frac{x}{2}\right)}{1 + \tan\left(\frac{x}{2}\right)} \]

### Step 5: Use Tangent Subtraction Formula
Recognize that this is the tangent subtraction formula:
\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
Here, \( a = \frac{\pi}{4} \) and \( b = \frac{x}{2} \):
\[ \frac{1 - \tan\left(\frac{x}{2}\right)}{1 + \tan\left(\frac{x}{2}\right)} = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) \]

Thus, we have:
\[ y = \tan^{-1}\left(\tan\left(\frac{\pi}{4} - \frac{x}{2}\right)\right) \]

### Step 6: Simplify the Inverse Tangent
Since \(\tan^{-1}(\tan \theta) = \theta\) for \(\theta\) in the principal range:
\[ y = \frac{\pi}{4} - \frac{x}{2} \]

### Step 7: Differentiate with Respect to \(x\)
Now, differentiate \( y = \frac{\pi}{4} - \frac{x}{2} \) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{\pi}{4} - \frac{x}{2}\right) \]
\[ \frac{dy}{dx} = 0 - \frac{1}{2} \]
\[ \frac{dy}{dx} = -\frac{1}{2} \]

### Final Answer
\[ \boxed{-\frac{1}{2}} \]