Express ` tan^(-1) ((cos x )/( 1- sin x) ) , ( -3 pi)/(2) lt x lt (pi)/(2)` in the simplest form

Answer

To express \( \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) \) in its simplest form for \( -\frac{3\pi}{2} < x < \frac{\pi}{2} \), we can 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
Recall the identity for tangent:
\[ \tan(y) = \frac{\sin(y)}{\cos(y)} \]
We can rewrite \( \frac{\cos x}{1 - \sin x} \) in terms of tangent:
\[ \tan(y) = \frac{\cos x}{1 - \sin x} \]

### Step 3: Multiply numerator and denominator by \( 1 + \sin x \)
To simplify, multiply both the numerator and denominator by \( 1 + \sin x \):
\[ \tan(y) = \frac{\cos x (1 + \sin x)}{(1 - \sin x)(1 + \sin x)} = \frac{\cos x (1 + \sin x)}{1 - \sin^2 x} = \frac{\cos x (1 + \sin x)}{\cos^2 x} \]

### Step 4: Simplify the expression
This simplifies to:
\[ \tan(y) = \frac{1 + \sin x}{\cos x} \]
Thus, we can express \( y \) as:
\[ y = \tan^{-1} \left( \frac{1 + \sin x}{\cos x} \right) \]

### Step 5: Recognize the tangent addition formula
Using the tangent addition formula:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \]
we can rewrite:
\[ y = \tan^{-1}(1) + \tan^{-1} \left( \frac{\sin x}{\cos x} \right) = \frac{\pi}{4} + \tan^{-1}(\tan x) \]

### Step 6: Simplify further
Since \( \tan^{-1}(\tan x) = x \) for \( -\frac{\pi}{2} < x < \frac{\pi}{2} \), we have:
\[ y = \frac{\pi}{4} + x \]

### Conclusion
Thus, the simplest form of the expression is:
\[ \tan^{-1} \left( \frac{\cos x}{1 - \sin x} \right) = x + \frac{\pi}{4} \]