If `(pi)/(2)lt thetalt(3pi)/(2),then sqrt((1-sintheta)/(1+sintheta))"is equal to"`

Options

A. `sec theta-tan theta`
B. `sec theta+tan theta`
C. `tan theta-sec theta`
D. none of these

Explanation

To solve the problem, we need to find the value of \(\sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}}\) given that \(\frac{\pi}{2} < \theta < \frac{3\pi}{2}\).

### Step-by-Step Solution:

1. **Start with the expression**:
\[ y = \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \]

2. **Rationalize the expression**:
Multiply the numerator and the denominator by \(\sqrt{1 - \sin \theta}\):
\[ y = \sqrt{\frac{(1 - \sin \theta)(1 - \sin \theta)}{(1 + \sin \theta)(1 - \sin \theta)}} \]
This simplifies to:
\[ y = \sqrt{\frac{(1 - \sin \theta)^2}{1 - \sin^2 \theta}} \]

3. **Use the Pythagorean identity**:
Recall that \(1 - \sin^2 \theta = \cos^2 \theta\):
\[ y = \sqrt{\frac{(1 - \sin \theta)^2}{\cos^2 \theta}} \]

4. **Simplify the square root**:
This can be simplified:
\[ y = \frac{1 - \sin \theta}{|\cos \theta|} \]
Since \(\frac{\pi}{2} < \theta < \frac{3\pi}{2}\), \(\cos \theta\) is negative. Therefore, \(|\cos \theta| = -\cos \theta\):
\[ y = \frac{1 - \sin \theta}{-\cos \theta} \]

5. **Separate the terms**:
This can be rewritten as:
\[ y = -\frac{1 - \sin \theta}{\cos \theta} = -\left(\frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta}\right) \]

6. **Express in terms of secant and tangent**:
Recognizing the trigonometric identities:
\[ y = -\sec \theta + \tan \theta \]
Rearranging gives:
\[ y = \tan \theta - \sec \theta \]

### Final Result:
Thus, the value of \(\sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}}\) is:
\[ \boxed{\tan \theta - \sec \theta} \]