`tan^-1[(cos x)/(1+sin x)]` is equal to

Options

A. `(pi)/(4) - (x)/(2)`, for `x in (-(pi)/(2), (3pi)/(2))`
B. `(pi)/(4) -(x)/(2), " for " x in (-(pi)/(2), (pi)/(2))`
C. `(pi)/(4), (x)/(2), " for " x in ((3pi)/(2), (5pi)/(2))`
D. `(pi)/(4) -(x)/(2), " for " x in (-(3pi)/(2), (pi)/(2))`

Explanation

To solve the expression \( \tan^{-1}\left(\frac{\cos x}{1 + \sin x}\right) \), we will follow these steps:

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

### Step 2: Use the identity for tangent
We can use the identity for tangent:
\[ \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) = \frac{\cos x}{1 + \sin x} \]
This identity is derived from the angle subtraction formula for tangent.

### Step 3: Set the equation
Since we have \( y = \tan^{-1}\left(\frac{\cos x}{1 + \sin x}\right) \), we can equate:
\[ y = \frac{\pi}{4} - \frac{x}{2} \]

### Step 4: Solve for \( y \)
Thus, we can express \( y \) as:
\[ y = \frac{\pi}{4} - \frac{x}{2} \]

### Step 5: Conclusion
Therefore, the expression \( \tan^{-1}\left(\frac{\cos x}{1 + \sin x}\right) \) simplifies to:
\[ \tan^{-1}\left(\frac{\cos x}{1 + \sin x}\right) = \frac{\pi}{4} - \frac{x}{2} \]