If `-pi lt theta lt -(pi)/(2)," then " |sqrt((1-sin theta)/(1+sintheta))+sqrt((1+sin theta)/(1-sin theta))|` is equal to :

Options

A. ` 2 sec theta`
B. ` -2 sec theta`
C. `2"sec"(theta)/(2)`
D. `-"sec"(theta)/(2)`

Answer

To solve the problem, we need to evaluate the expression

\[ \left| \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} + \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \right| \]

for the range \(-\pi < \theta < -\frac{\pi}{2}\).

### Step 1: Write the expression
We start with the expression:

\[ E = \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} + \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \]

### Step 2: Find a common denominator
To combine the two square root terms, we find a common denominator:

\[ E = \frac{\sqrt{(1 - \sin \theta)^2} + \sqrt{(1 + \sin \theta)^2}}{\sqrt{(1 + \sin \theta)(1 - \sin \theta)}} \]

### Step 3: Simplify the numerator
The numerator simplifies to:

\[ \sqrt{(1 - \sin \theta)^2} + \sqrt{(1 + \sin \theta)^2} = |1 - \sin \theta| + |1 + \sin \theta| \]

### Step 4: Simplify the denominator
The denominator can be simplified using the identity \(1 - \sin^2 \theta = \cos^2 \theta\):

\[ \sqrt{(1 + \sin \theta)(1 - \sin \theta)} = \sqrt{1 - \sin^2 \theta} = \cos \theta \]

### Step 5: Substitute back into the expression
Now we can rewrite \(E\):

\[ E = \frac{|1 - \sin \theta| + |1 + \sin \theta|}{\cos \theta} \]

### Step 6: Determine the signs of the terms
For \(-\pi < \theta < -\frac{\pi}{2}\), we know that \(\sin \theta\) is negative. Therefore:

- \(1 - \sin \theta\) is positive (since \(\sin \theta < 0\))
- \(1 + \sin \theta\) is also positive (since \(\sin \theta > -1\))

Thus, we can drop the absolute values:

\[ E = \frac{(1 - \sin \theta) + (1 + \sin \theta)}{\cos \theta} = \frac{2}{\cos \theta} \]

### Step 7: Final expression
Now we have:

\[ E = \frac{2}{\cos \theta} \]

### Step 8: Apply the modulus
Since \(\cos \theta\) is negative in this range, we take the modulus:

\[ |E| = \left| \frac{2}{\cos \theta} \right| = -\frac{2}{\cos \theta} \]

### Step 9: Final result
Thus, the final expression simplifies to:

\[ |E| = -\frac{2}{\cos \theta} \]

### Step 10: Relate to sine
Using the identity \(\cos \theta = -\sin\theta\) in this range, we get:

\[ |E| = -\frac{2}{-\sin \theta} = \frac{2}{\sin \theta} \]

### Conclusion
So, the final answer is:

\[ |E| = -2 \sin \theta \]